ICM 2006

Short Communications Abstracts Section 14 Combinatorics ICM 2006 – Short Communications. Abstracts. Section 14

Reciprocal graphs

G. Indulal, A. Vijayakumar*

Department of Mathematics, Cochin University of Science and Technology, Cochin-682 022, India [email protected]

Eigenvalue of a graph is the eigenvalue of its adjacency matrix. A graph G is reciprocal if the reciprocal of each of its eigenvalue is also an eigenvalue 1 P of G. The Wiener index W (G)of a graph G is defined by W (G) = 2 d∈D d r Q and the Multiplicative Wiener index Wπ(G) by Wπ(G) = d where d6=0∈D D is the distance matrix of G.In this paper, some classes of reciprocal graphs ,an upperbound for their energy,a pair of equi energetic reciprocal graphs on every n ≡ 0 mod (12) and n ≡ 0 mod (16), the Wiener indices and Multiplicative Wiener indices of some classes of reciprocal graphs are discussed.

References

[1] D.M.Cvetkovi´c,Graphs with least eigenvalue -2; a historical survey and recent developments in maximal exceptional graphs. Linear Algebra Appl. 356 (2002) 189-210. [2] J.R.Dias,Properties and relationships of right-hand mirror-plane fragments and their eigenvectors : the concept of complementarity of molecular graphs,Molecular Physics, 1996, Vo l . 88, No . 2, 407- 417 [3] I. Gutman, Topology and stability of conjugated hydrocarbons. The depen- dence of total pi-electron energy on molecular topology, J. Serb. Chem. Soc. 70 (2005) 441-456. [4] I.Gutman,W.Linert,I.Lukovits and Z.Tomovi´c,On the Multiplicative Wiener Index and Its Possible Chemical Applications,Monatshefte f¨urChemie 131,421- 427(2000) [5] G. Indulal, A. Vijayakumar: On a Pair of Equienergetic Graphs,(MATCH)Volume 55 (2006) 83-90.

ICM 2006 – Madrid, 22-30 August 2006 1 ICM 2006 – Short Communications. Abstracts. Section 14

Spiral chains: A new proof of the four color theorem

I. Cahit Faculty of Engineering, Cyprus International University, Lefkose, North Cyprus [email protected]

2000 Mathematics Subject Classification. 05C Acceptable, but due to extensive usage of a computer rather unpleasant proof of the famous four color map problem of Francis Guthrie were settled eventually by W. Appel and K. Haken in 1976 [1]. Using the same method but shortening the proof twenty years later by another team, namely N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas would not im- prove considerably the readability of the proof either [2]. Thus it has been widely accepted the need of more elegant and readable proof. There are considerable number of equivalent formulations of the problem but none of them promising for a possible non-computer proofs. On the other hand known proofs are used the concept of Kempe chain and reducibility of the configurations which were a century old ideas [3]. With these in mind we have introduced a new concept which we call “spiral chains” in the max- imal planar graphs and instead of sticking to the reducible configurations which were the main complexity issue in the classical proof, we prefer to show the four colorability of the spiral chains by starting coloring from the inner vertices of a maximal planar graphs [4]. We have shown that for any maximal planar graph as long as spiral chains are being used we do not need the fifth color. Finally by the use of spiral chain coloring we have also given an independent proof of Tait’s conjecture that every (bridgeless) cu- bic planar graph has a three -edge -coloring [5]. Henceforth this paper offers another proof of the four color theorem in two ways which is not based on deep and abstract theories from the other branches of mathematics or using computing power of computers, but rather completely on a new idea in graph theory.

References

[1] K. Appel and W. Haken, “Every planar map is four colorable”, Contemporary Math., 98 (1989). [2] N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas, “The four colour theorem”, J. Combin. Theory Ser.,B. 70 (1997), 2-44. [3] A. B. Kempe, “On the geographical problem of the four colours”, American J. of Math., 2(3),193-200, 1879.

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[4] I. Cahit, “Spiral chains: A new proof of the four color theorem”, arXiv preprint, math. CO/0408247, August 18, 2004. [5] I. Cahit, ”Spiral chains: The proofs of Tait’s and Tutte’s three-edge-coloring conjectures”, arXiv preprint, math. CO/0507127 v1, July 6, 2005.

ICM 2006 – Madrid, 22-30 August 2006 3 ICM 2006 – Short Communications. Abstracts. Section 14

Green’s function and Poisson kernel on a path

E. Bendito, A. Carmona*, A. M. Encinas, J. M. Gesto

MAIII, UPC, Jordi Girona Salgado 1-3, 08034 Barcelona, Spain [email protected]

2000 Mathematics Subject Classification. 39A Green’s function and Poisson kernel of networks can be used to study dif- fusion problems on graphs, such as hitting time, chip-firing and discrete Markov chains. The reason is that these problems are written in terms of boundary value problems with respect to the Laplace operator associated with the network and the Green’s function or the Poisson kernel are the re- solvent kernels. Therefore, Green’s functions and Poisson kernels constitute a powerful tool in dealing with a wide range of combinatorial problems, see [2, 3]. The authors studied in [2] general boundary value problems on arbitrary subsets of a network and in particular they obtained a relation between the Green’s function and the Poisson kernel. In general it is not easy to ob- tain and explicit expression for that functions. However, when the network presents symmetries they can be obtained by hand. Our objective here is to obtain explicit expressions for the Green func- tion and Poisson kernel for the Sturm-Liouville problem associated with the Schr¨odinger operator on the path Pn = {0, . . . , n + 1}. In this case the explicit construction of such functions is a mechanic process since one has two independent solutions of the homogeneous problem associated with the differences equation. Specifically, they are given through Chebyshev Poly- nomials, since they verified a recurrence relation of the type Pi+2(x) = 2 x Pi+1(x) − Pi(x), i ∈ Z. For instance, if we consider the Neumann regular problem, that is,

2(1+q)z(i)−z(i+1)−z(i) = f(i) , i = 1, . . . , n, z(0)−z(1) = z(n+1)−z(n) = 0, where q > 0, then the Green function is given by  V (1 + q) V (1 + q) , 0 ≤ k ≤ s ≤ n, 1  k−1 n−s Gq(k, s) = 2q Un−1 (1 + q)   Vs−1 (1 + q) Vn−k (1 + q) , 1 ≤ s ≤ k ≤ n + 1, where Uk and Vk denote the Second and Third kind Chebyshev polynomials, respectively, [1].

4 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

References

[1] Mason, J.C. and Handscomb, D.C., Chebyshev Polynomials. Chapman & Hall, CRC Press Company, 2003. [2] Bendito, E., Carmona, A., Encinas, A.M., Solving boundary value problems on networks using equilibrium measures, J. Funct. Anal. 17 (2000), 155–176. [3] Chung, F.R.K., Yau, S.T., Discrete Green functions, J. Comb. Theory 91 (2000), 191–214.

ICM 2006 – Madrid, 22-30 August 2006 5 ICM 2006 – Short Communications. Abstracts. Section 14

Independent arcs of acyclic orientations of complete r-partite graphs

Gerard Jennhwa Chang*, Chen-Ying Lin‡ and Li-Da Tong‡

Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan; ‡Department of Applied Mathematics, National Sun Yet-sen University, 804, Taiwan [email protected]

2000 Mathematics Subject Classification. 05C Suppose D is an acyclic orientation of a graph G. An arc of D is said to be independent if its reversal results another acyclic orientation. Denote i(D) the number of independent arcs in D and N(G) = {i(D): D is an acyclic orientation of G}. Also, let imin(G) be the minimum of N(G) and imax(G) the maximum. While it is known that imin(G) = |V (G)|−1 for any connected graph G, the present paper determines imax(G) for complete r- partite graphs G. We also determine N(G) for balanced complete r-partite graphs G, and find some complete r-partite graphs G whose N(G) is a set of consecutive integers. As a consequence, we answer a question of West that there exist graphs G whose N(G) are not a set of consecutive integers.

References

[1] K. L. Collins and K. Tysdal, Dependent edges in Mycielski graphs and 4- colorings of 4-skeletons, J. Graph Theory 46 (2004), 285–296. [2] D. C. Fisher, K. Fraughnaugh L. Langley, and D. B. West, The number of dependent arcs in an acyclic orientation, J. Combin. Theory Ser. B 71 (1997), 73–78. [3] K.-W. Lih, C.-Y. Lin and L.-D. Tong, On an interpolation property of outer- planar graphs, Discrete Applied Math. 154 (2006), 166–172. [4] K. Tysdal, Dependent Edges in Acyclic Orientations of Graphs, Ph.D. Thesis, Wesleyan University, Middletown, CT, May 2001. [5] D. B. West, Acyclic orientations of complete bipartite graphs, Discrete Math. 138 (1995), 393–396.

6 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

Cops, Robber, and Alarms

Nancy E. Clarke*, Emma L. Connon

Department of Mathematics and Statistics, Acadia University, Wolfville, Nova Scotia, Canada; Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada [email protected]

2000 Mathematics Subject Classification. 05C75, 05C99, 91A43 The two games considered are mixtures of Searching and Cops and Robber. The cops have partial information, provided first via selected vertices of a graph, and then via selected edges. The robber has perfect information. In [1, 3], this partial information includes both the robber’s position and the direction in which he is moving. Bounds are given on the amount of such information required by a single cop to guarantee the capture of the robber on a tree. These bounds are then generalized to copwin graphs, using the notion of copwin spanning tree [2]. In this paper, the partial information includes the robber’s position only, and there is no directional signal available to the cops. These variations of the game were first introduced in [1], where bounds are given on the amount of such information required by one cop to guarantee the capture of the rob- ber on a tree. Unfortunately, because of the lack of directional information, these bounds cannot be generalized to copwin graphs via copwin spanning trees. In this paper, we take a modified approach and we give bounds on the amount of partial information required by one cop to guarantee the capture of the robber on an arbitrary copwin graph. We then compare our results with the tree results presented in [1].

References

[1] Clarke, N.E., A Game of Cops and Robber Played with Partial Information, Congressus Numerantium 166 (2004), 145-159. [2] Clarke, N.E. and Nowakowski, R.J., Cops, Robber, and Traps, Utilitas Math- ematica 60 (2001), 91-98. [3] Clarke, N.E. and Nowakowski, R.J., Cops, Robber, and Photo Radar, Ars Com- binatoria 56 (2000), 97-103.

ICM 2006 – Madrid, 22-30 August 2006 7 ICM 2006 – Short Communications. Abstracts. Section 14

Semifields and semifield planes

Minerva Cordero

Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019, USA [email protected]

2000 Mathematics Subject Classification. Primary 51E15, 51A40; Secondary 05B25, 17A35 A semifield is a not necessarily associative division ring; a semifield plane is an affine plane coordinatized by a semifield. The study of finite semi- fields was originated by L.E. Dickson in 1905. In the late 50’s and early 60’s some new classes were constructed. It was conjectured that semifields were much more plentiful than quasifields (semifields minus distributivity). However since 1965 only a few classes of semifields have been constructed. Yet the techniques of derivation and homology replacement discovered by Ostrom [4] have given a profusion of nonisomorphic quasifields. A survey of all the known semifield planes up to 1965 was given in [3]. In 1999 an updated account was given in [1]. Since then several new classes have been constructed, e.g. by Kantor [2] and Prince [5]. In this talk we present a survey of the semifields constructed in the last decade and introduce a new class recently constructed by the author.

References

[1] M. Cordero and G. P. Wene, A survey of finite semifields,Discrete Math., 208/209, 1999, 125–137. [2] W. M. Kantor, Commutative semifelds and symplectic spreads, J. Algebra 270 (2003), 96–114. [3] D.E. Knuth, Finite semifields and projective planes,J. Algebra 2 (1965), 182– 217. [4] T.G. Ostrom, Finite Translation Planes, Lecture Notes in Mathematics, 158, Springer, New York, 1970. [5] A. R. Prince, Two new families of commutative semifields, Bull. LMS 32 (2000), 547–550.

8 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

Star complements and exceptional graphs

D. Cvetkovi´c*a, P. Rowlinsonb, S. K. Simi´ca aDepartment of Mathematics, Faculty of Electrical Engineering, University of Belgrade, P.O. Box 35-54, 11120 Belgrade, Serbia and Montenegro; bDepartment of Computing Science and Mathematics, University of Stirling, Stirling FK9 4LA, Scotland, U.K.

2000 Mathematics Subject Classification. 05C50 Let G be a finite graph of order n with an eigenvalue µ of multiplicity k. (Thus the µ-eigenspace of a (0, 1)-adjacency matrix of G has dimension k.) A star complement for µ in G is an induced subgraph G − X of G such that |X| = k and G−X does not have µ as an eigenvalue. An exceptional graph is a connected graph, other than a generalized line graph, with least eigenvalue greater than or equal to −2. Prompted by the study of graphs with least eigenvalue −2 [1], we establish some properties of star complements [3], and of eigenvectors, of exceptional graphs with least eigenvalue −2. Star complements appear as the graphs with least eigenvalue greater than −2 which have been characterized in [2]. In particular, we study one vertex extensions of exceptional graphs with least eigenvalue greater than −2. We also discuss integral eigenvectors of exceptional graphs having −2 as a simple eigenvalue.

References

[1] Cvetkovi´c D., Rowlinson P., Simi´c S. K., Spectral Generalizations of Line Graphs: On Graphs with Least Eigenvalue −2, Cambridge University Press, Cambridge, 2004. [2] Doob M., Cvetkovi´cD., On spectral characterizations and embedding of graphs, Linear Algebra Appl. 27 (1979), 17-26. [3] Rowlinson P., Star complements in finite graphs: a survey, Rendiconti Sem. Mat. Messina 8 (2002), 145-162.

ICM 2006 – Madrid, 22-30 August 2006 9 ICM 2006 – Short Communications. Abstracts. Section 14

New worst-case upper bound for counting independent sets

De Ita Guillermo*, Erica Vera

Fac. Cs. de la Computaci´on,Universidad Autonoma de Puebla, M´exico [email protected]

2000 Mathematics Subject Classification.05A06 Different algorithms are presented to carry out the exact counting of the number of independent sets of a graph G, denoted by NI(G). To compute NI(G) is a classic #P-complete problem for graphs of degree 3 or greater [1]. With regard to graphs of degree 3, we establish, based on the topological structure of the graph G, a worst-case upper bound for computing NI(G) of O(poly(n)∗1.1892n) and being n the number of nodes of the graph. We also determine new polynomial classes for the counting problem of computing NI(G). Applying relations found between Fibonacci series and the number of independent sets of a graph G, we show that if the depth-first search of the graph builds a free tree and a set of fundamental non-intersected cycles, that is, there are not common edges among none of any two pairs of those fundamental cycles, then the problem NI(G) is tractable. The new polynomial class for NI(G) does not put restrictions on the degree of the graph, but rather it depends on the topological structure of the graph. The exact method showed here could be used to impact directly to speed up many other algorithms for counting problems, and to influ- ence on the complexity time of those algorithms, i.e. for counting models in propositional Boolean formulas, counting colorings of graphs, counting exact covers, etc.

References

[1] Greenhill Catherine , The complexity of counting colourings and independent sets in sparse graphs and hypergraphs”, Computational Complexity, 1999. [2] Dahll¨ofV., Jonsonn P., Wahlstr¨omM., Counting models for 2SAT and 3SAT formulae., Theoretical Computer Sciences, 332(1-3), 2005, 265-291. [3] Dyer M., Greenhill C., Some #P-completeness Proofs for Coulorings and In- dependent Sets, Research Report Series, University of Leeds, 1997. [4] Roth D., On the hardness of approximate reasoning, Artificial Intelligence 82, (1996), 273-302.

10 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

Some results about the totally positive completion problem

Ramad´anEl-Ghamry ?, Cristina Jord´an,Juan R. Torregrosa Dpto. Matem´atica Aplicada, Universidad Polit´ecnica de Valencia, Spain el [email protected]; [email protected]; [email protected]

2000 Mathematics Subject Classification. 15A15, 15A48

A partial matrix over R is an n × n array A = (aij) in which some entries are specified real numbers while the remaining (unspecified) entries are free to be chosen from R. A partial matrix is said to be combinatorially sym- metric when the (i, j) entry is specified if and only if the (j, i) entry is, and is non-combinatorially symmetric in the other case. A completion of a partial matrix is a choice of values for the unspecified entries resulting a conventional matrix. A matrix completion problem asks which partial ma- trices have a completion with some desired property. The specified positions in an n × n non-combinatorially symmetric partial matrix A = (aij) can be represented by a directed graph GA = (V,E), where the set of vertices V is {1, 2, . . . , n} and (i, j), i 6= j, is an arc of E if the (i, j) entry is specified. An n × n real matrix A = (aij) is said to be totally positive (nonneg- ative) if every minor is positive (nonnegative). These matrices are having an increasing importance in approximation theory, combinatorics, statistic, economics, computer aided geometric design and wavelets, etc. Total nonnegativity is inherited by submatrices. Therefore, it is a neces- sary condition that every fully specified submatrix be totally nonnegative. A partial matrix that satisfied this necessary condition is said to be par- tial totally nonnegative. Here we are interested in the totally nonnegative completion problem, that is, when a partial totally nonnegative matrix has a totally nonnegative completion. This problem was studied for combina- torially symmetric partial matrices by Johnson, Kroschel and Lundquist in [1] and by Jord´anand Torregrosa in [2], and by el-Ghamry, Jord´anand Torregrosa for the non-combinatorially symmetric case in [3]. In this work we completed the study done in [3]. We get necessary and sufficient conditions in order to obtain a totally nonnegative completion of a partial totally nonnegative matrix, whose associated graph is a path, a cycle, a totally specified path, a double-path, etc.

References

[1] Johnson, C.R., Kroschel B.K. and Lundquist M., The Totally Nonnegative Completion Problem, Fields Institute Commnunications, American Mathemat- ical Society, Providence, RI. 18 (1998), 97–109.

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[2] Jord´an,C. and Torregrosa, Juan R., The totally positive completion problem. Linear Algebra and its Applications 393 (2004), 259–274. [3] el-Ghamry, R., Jord´an,C. and Torregrosa, Juan R., The completable digraphs for the totally positive completion problem, Discrete Applied Mathematics. To appear.

12 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

A bijection between 2-triangulations and pairs of non-crossing Dyck paths

Sergi Elizalde

Department of Mathematics, Dartmouth College, 6188 Bradley Hall, Hanover, NH 03755, USA [email protected]

2000 Mathematics Subject Classification. 05A15, 52B05 It is well known that the number of triangulations of a convex n-gon is 1 2(n−2)
the Catalan number Cn−2 = n−1 n−2 . A natural generalization of a triangulation is a k-triangulation, which is defined to be a maximal set of diagonals so that no k + 1 of them mutually cross in their interiors. For example, a 1-triangulation is just a triangulation in the standard sense. It was recently proved by Jonsson [2] that, remarkably, the number of k- triangulations of an n-gon is given by the following determinant of Catalan numbers:

C C ...C C n−2 n−3 n−k n−k−1 C C ...C C k n−3 n−4 n−k−1 n−k−2 det(Cn−i−j) = . . . . . i,j=1 ......

Cn−k−1 Cn−k−2 ...Cn−2k+1 Cn−2k

Using the Lindstr¨om-Gessel-Viennot method [1], this determinant is also known to enumerate k-tuples (P1,P2,...,Pk) of Dyck paths of semilength n − 2k such that each Pi never goes below Pi+1. In the case k = 1, this determinant is just Cn−2, which counts Dyck paths from of semilength n−2. There are several simple bijections between triangulations of a convex n-gon and such paths [4]. One of the main open questions left in [2], stated also in [3, Problem 1], is to find a bijection between k-triangulations and k-tuples of non-crossing Dyck paths, for k ≥ 2. Here we solve this problem for k = 2, that is, we present a bijection between 2-triangulations of a convex n-gon and pairs (P,Q) of Dyck paths from of semilength n−4 so that P never goes below Q. The bijection is obtained by constructing isomorphic generating trees for the sets of 2-triangulations and pairs of non-crossing Dyck paths. We also discuss possible generalizations of our work, including a gen- erating tree for k-triangulations for arbitrary k, which may lead to solving the open problem of finding an analogous bijection for k ≥ 3.

ICM 2006 – Madrid, 22-30 August 2006 13 ICM 2006 – Short Communications. Abstracts. Section 14

References

[1] Gessel, I., Viennot, G., Binomial determinants, paths, and hook length formu- lae, Adv. Math. 58 (1985), 300–321. [2] Jonsson, J., Generalized triangulations and diagonal-free subsets of stack poly- ominoes, J. Combin. Theory Ser. A 112 (2005), 117–142. [3] C. Krattenthaler, Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes, preprint, arxiv:math.CO/0510676. [4] R.P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge University Press, Cambridge, 1999.

14 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

Binomial coefficients in equalities for natural numbers

I. Erusalimskiy

Department of Mechanics and Mathematics, Rostov State University, st. Bolshaya Sadovaya, 105, Rostov-on-Don, 344007 Russia [email protected]

2000 Mathematics Subject Classification. 05A10 There is a well-known [1, p. 331] formula for quantity of surjective functions from a set X with m elements to a set Y with n elements:

X m 1 m 2 m 3 m n−1 n−1 m surY = n −Cn(n−1) +Cn(n−2) −Cn(n−3) +···+(−1) Cn ·1 (1) Analysis of the proof of this statement shows that it’s valid for any m and n. In the case when 1 ≤ m < n the set surY X = and we obtain the next equalities:

1 2 3 n−2 n−1 n = Cn(n − 1) − Cn(n − 2) + Cn(n − 3) − · · · + (−1) Cn · 1, 2 1 2 2 2 3 3 n−2 n−1 2 n = Cn(n − 1) − Cn(n − 2) + Cn(n − 3) − · · · + (−1) Cn · 1 , ········· (2) n−1 1 n−1 2 n−1 3 n−1 n−2 n−1 n−1 n = Cn(n−1) −Cn(n−2) +Cn(n−3) −· · ·+(−1) Cn ·1 . These equalities are interested because they give linear representation of natural number and its degrees from smaller natural numbers and their degrees with the same binomial coefficients. For example, in case n = 5 we have:

5 = 5 · 4 − 10 · 3 + 10 · 2 − 5 · 1; 52 = 5 · 42 − 10 · 32 + 10 · 22 − 5 · 12; 53 = 5 · 43 − 10 · 33 + 10 · 23 − 5 · 13; 54 = 5 · 44 − 10 · 34 + 10 · 24 − 5 · 14;

If m = n when it is known that each surjective function appears to   be bijective function surY X = biY X and we obtain from (1) the next equality:

n 1 n 2 n 3 n n−1 n−1 n! = n − Cn(n − 1) + Cn(n − 2) − Cn(n − 3) + ··· + (−1) Cn . (3)

We named this equality as “additive representation for n!” ([2]).

ICM 2006 – Madrid, 22-30 August 2006 15 ICM 2006 – Short Communications. Abstracts. Section 14

The equalities below follow from equalities (2):

q  1 p 2 p 3 p n−2 n−1 p Cn(n − 1) − Cn(n − 2) + Cn(n − 3) − · · · + (−1) Cn · 1 = 1 p·q 2 p·q 3 p·q n−2 n−1 p·q = Cn(n − 1) − Cn(n − 2) + Cn(n − 3) − · · · + (−1) Cn · 1 , (4) 1 ≤ p · q < n, p, q ∈ N.

These results will be obtained “automatically” and author doesn’t know another method for it’s proving.

References

[1] Rosen, Kenneth H. Discrete Mathematics and Its Applications. McGraw-Hill, Inc. 2nd ed., ISBN 0–07–053744–5. [2] Erusalimskyi I.Discrete Mathematics: Theory, Exercises and Applications (in Russian) Moskow, Vuzovskaya kniga, 2005, 7 ed., 268 pp. ISBN 5–9502–0123– X.

16 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

On the eigensharp and almost eigensharp graphs

E. Ghorbani*, H. R. Maimani

Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran e [email protected]

2000 Mathematics Subject Classification. 05C50, 15A18, 05C70 The minimum number of complete bipartite subgraphs needed to partition the edges of a graph G is denoted by b(G). A known cover bound on b(G) states that b(G) ≥ max {p(G), q(G)}, where p(G) and q(G) are the numbers of positive and negative eigenvalues of G, respectively. If the equality holds, then G is called eigensharp and if b(G) =max{p(G), q(G)} + 1, then G is called almost eigensharp. In this paper we show that the graphs with at most one cycle are eigensharp and investigate the eignsharpness of the products of some families of graphs. It is also proved that n-cubes (n odd) and n-wheels (n ≡ 0, 3 (mod 4)) are eigensharp. Finally, we obtain some results on almost eigensharp graphs.

References

[1] D. Cvetkovi´c,P. Rowlinson, S. Simi´c, Eigenspaces of Graphs, Cambridge Univ. Press, 1997. [2] R. L. Graham, H. O. Pollak, On the addressing problem for loop switching, J. Bell System Tech. 50 (1971), 2495–2519. [3] D. A. Gregory, B. L. Shader, V. L. Watts, Biclique decomposition and hermitian rank, Linear Algebra Appl. 292 (1999), 267–280. [4] T. Kratzke, B. Reznick, D. West, Eigensharp graphs: decomposition into com- plete bipartite subgrahs, Trans. Amer. Math. Soc. 308 (1988), 637–653. [5] S. D. Monson, N. J. Pullman, R. Ress, A survey of clique and biclique covering and factorization of (0-1) matrices, Bull. I.C.A. 14 (1995), 17–86.

ICM 2006 – Madrid, 22-30 August 2006 17 ICM 2006 – Short Communications. Abstracts. Section 14

Labeled directed graphs and applications

S. M. Hegde

Dept. of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Srinivasnagar-575025, INDIA. [email protected]

2000 Mathematics Subject Classification. 05C78 Graph labelings, where the vertices and edges are assigned, real values sub- ject to certain conditions, have often been motivated by their utility to various applied fields and their intrinsic mathematical interest (logico – mathematical). Labeled graphs are becoming an increasingly useful family of Mathematical Models for a broad range of applications. While the qual- itative labelings of graph elements have inspired research in diverse fields of human enquiry such as Conflict resolution in social psychology electrical circuit theory and energy crisis, etc., quantitative labelings of graphs have led to quite intricate fields of application such as Coding Theory problems, in determining ambiguities in X-Ray Crystallographic analysis, to Design Communication Network addressing Systems, in determining Optimal Cir- cuit Layouts and Radio- Astronomy., etc. A directed graph D with n vertexs ands e arcs, no self-loops and mul- tiple edges is labeled by assigning to each vertex a distinct element from the set Ze+1 = {0,1,2,. . . ,e}. An arc (x, y) from vertex x to y is labeled with λ(xy)= (λ(x) − λ (y))mod (e+1), where λ(x) and λ(y) are the values assigned to the vertices x and y. Such a labeling is a graceful labeling of D if all the λ(xy) are distinct. Then D is called a graceful digraph. In this short communication, we present graceful labelings of directed graphs and their applications to sequenciable cyclic groups, complete mappings and neofields, i.e., (i)The unidirectional path is graceful iff Zn is sequence- able, (ii) A graceful labeling of the Collections of unicycles and paths P P (∪Cki) ∪ (∪Phj)(i=1,2,. . . ,r, j=1,2,. . . ,s), where ki + hj= n = e+s, occurs if and only if there exists a (K,1) near complete mapping of Zn = Ze+s, where K ={ k1,k2,. . . ,kr;h1,h2,. . . ,hs}and (iii) Let Hn be a cyclic group of order n and Nn+1= Hn∪{0}. Then (Nn+1,+,*) is a cyclic neofield if and only if the digraph D with specified property is graceful.

References

[1] Bloom, G. S., and Hsu, D. F., On graceful directed graphs, SIAM J. Alg. Disc. Math., vol. 6,.no. 3, July 1985, 519-536.

18 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

[2] Gallian, J. A., A dynamic survey of graph labeling, Electronic J. of Combina- torics, DS#6 , 2005, 1-148.

ICM 2006 – Madrid, 22-30 August 2006 19 ICM 2006 – Short Communications. Abstracts. Section 14

On commuting graph of finite groups

A. Iranmanesh*, A. Jafarzadeh

Department of Mathematics, Tarbiat Modarres University, P.O.Box: 14115-137, Tehran, Iran [email protected]

2000 Mathematics Subject Classification. 05C25 Commuting graph of a group G, denoted by Γ(G), is a simple undirected graph whose vertices are all non-central elements of G and two distinct vertices x, y are adjacent if xy = yx. Commuting graph of a subset of a group is defined similarly. We denote the symmetric group and the alternating group on n let- ters, the dihedral group of order 2n and the Dicyclic group of order 4n by Sn,An, Dn and DCn respectively. Also Tn and In denote the sets of all transpositions and involutions in Sn, respectively. In this paper, we find the necessary and sufficient conditions for the connectivity of the graphs Γ(Sn), Γ(An), Γ(Tn), Γ(In), Γ(Dn) and Γ(DCn), and find their diameter or in some cases, a good upper bound for their diameter, when connected. Also we find the number of their components and try to find maximum and minimum degrees and clique and independence numbers for these graphs.

References

[1] Abdollahi, A., Akbari, S., Maimani, H. R., Non-commuting graph of a group, to appear in J. Algebra. [2] Moghaddamfar, A. R., Shi, W. J., Zhou, W., Zokayi, A.R., On the noncom- muting graph associated with a finite group, Sib. Math. J. 46 (2005), 325–332.

[3] Segev, Y., Seitz, G. M., Anisitropic groups of type An and the commuting graph of finite simple groups, Pacific J. Math. 202 (2002), 125–225.

20 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

An analogue of Tutte’S 5-flow conjecture for undirected graphs

G. B. Khosrovshahi

School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics(IPM) and University of Tehran, P. O. Box 19395-5746, Tehran, Iran [email protected]

2000 Mathematics Subject Classification. 05C20 Let G be a graph and k be a positive integer. We say that G admits a k-flow if there exist an orientation of G and an assignment of weights from the set {0, ±1, ±2,..., ±(k − 1)} to the edges of G such that the sum of the weights of incoming edges and the sum of the weights of outgoing edges are equal. In 1954 Tutte made the following conjecture: Every bridgeless graph has a nowhere zero 5-flow. In 1981 Seymour [1] proved that the graphs with no cut edge has a nowhere zero 6-flow. In this talk we discuss an analogue of the conjecture for undirected graphs. A zero sum flow of an undirected graph G is an assignment which assigns a real number to each edge of G such that the sum of the assignments of edges adjacent to every vertex is zero. Now we conjecture that if G admits a nowhere zero zero sum flow, then G admits a nowhere zero zero sum 5-flow. Among some results we show the correctness of the conjecture for r-regular graphs(r ≥ 3).

References

[1] Seymour, P. D., Nowhere-zero 6-flows, J. Combin. Theory Ser. B 30 (1981), 130–135. [2] Steinberg, R., Tutte’s 5-flow conjecture for the projective plane, J. Graph The- ory 8 (1984), 277–289.

ICM 2006 – Madrid, 22-30 August 2006 21 ICM 2006 – Short Communications. Abstracts. Section 14

Recent results on semiovals

Gy. Kiss

Department of Geometry, E¨otv¨osLor´and University, 1117 Budapest, P´azm´any s 1/c, Hungary and Bolyai Institute, University of Szeged, 6720 Szeged, Aradi v´ertan´uktere 1, Hungary [email protected]

2000 Mathematics Subject Classification. 51E21

A semioval in a projective plane Πq of order q is a non-empty pointset S with the property that for every point P in S there exists a unique line tP such that S ∩ tp = {P }. This line is called the tangent to S at P . The classical examples of semiovals arise from polarities (ovals and unitals), and from the theory of blocking sets (the vertexless triangle). The study of semiovals is motivated by their applications to cryptography [1]. The aim of this talk is to investigate semiovals which are contained in the union of three lines but are not contained in the union any two of these lines. A complete classification of semiovals in PG(2, q) which are conntained in the sides of a triangle is given [3]. If the three lines are concurrent, then there are only two known examples. First, an infinite family arising from Baer subplanes of PG(2, q), where q is an even power of a prime. And second, a sporadic example in PG(2, 5), where S is an irreducible conic and the intersection of the three lines is any inner point of it. The semiovals of the above infinite family have an additional property, we called these semiovals strong ones. We present some necessary conditions for the existence of such objects and give a complete classification of strong semiovals in PG(2, p) and PG(2, p2), p an odd prime [2].

References

[1] Batten, L. M., Determining sets, Australas. J. Combin. 22 (2000), 167–176. [2] Blokhuis, A., Kiss, Gy., Malniˇc,A., Maruˇsiˇc,D., Kov´acs, I. and Ruff, J., Semio- vals contained in the union of three concurrent lines, J. Combin. Designs, sub- mitted. [3] Kiss, Gy. and Ruff, J., Notes on small semiovals, Ann. Univ. Sci. Budapest. E¨otv¨osSect. Math. 47 (2004), 97–105.

22 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

Bell numbers—unordered and ordered—and algebraic differential equations

Martin Klazar

Department of Applied Mathematics, Charles University, Malostransk´en´am. 25, Praha 118 00, Czech Republic [email protected]

2000 Mathematics Subject Classification. 11B73 (12H20)

Unordered Bell numbers bn count partitions of [n] = {1, 2, . . . , n} in nonempty disjoint sets and ordered Bell numbers cn count the mappings from [n] to [n] whose image is [m], m ≤ n. It is well known that their exponential generating functions are b xn c xn 1 X n = exp(exp(x) − 1) and X n = . n! n! 2 − exp(x) n≥0 n≥0

These power series satisfy algebraic differential equations (ADE) but in [1] it was proved that the ordinary generating function

X n B(x) = bnx n≥0 satisfies no ADE over C(x); the proof is based on the functional equation P n B(x/(1 + x)) = 1 + xB(x). It is not widely known that C(x) = cnx 1 1 satisfies a similar identity too, C(x/(1 + x)) = 2 + (x + 2 )C(x). In the talk we will discuss attempts to prove, using this identity, that C(x) satisfies no ADE either and we will consider a relation between functional equation P n P n satisfied by anx and differential equation satisfied by anx /n!.

References

[1] Klazar, M., Bell numbers, their relatives, and algebraic differential equations, J. Combin. Theory Ser. A 102 (2003), 63–87.

ICM 2006 – Madrid, 22-30 August 2006 23 ICM 2006 – Short Communications. Abstracts. Section 14

Colorings of quadrangulations of the torus and the Klein bottle

Daniel Kr´al’

Institute for Theoretical Computer Science, Charles University, Malostransk´e n´amˇest´ı25, 118 00 Prague, Czech Republic [email protected]

2000 Mathematics Subject Classification. 05C15, 05C10 Motivated by a question of Thomassen, we show that a triangle-free quad- rangulation of the torus is 3-colorable if and only if it does not contain the Cayley graph for the group Z13 and the set {1, 5} as a subgraph. Our result yields a proof of the conjecture of Archdeacon et al. [1] that every quadrangulation of the torus with representativity at least six is 3-colorable. We also show that every triangle-free quadrangulation of the Klein bot- tle with even meridian walk that does not contain a non-contractible sep- arating 4-cycle is 3-colorable. This yields a proof of another conjecture of Archdeacon et al. [1] that every quadrangulation of the Klein with repre- sentativity at least five is 3-colorable. In the proofs of both of our results, we first establish that a minimal counterexample must be a 4-regular quadrangulation of the surface and we then apply the characterization of such quadrangulations by Thomassen [2]. (The talk is based on a joint work with Robin Thomas.)

References

[1] Archdeacon, D., Hutchinson, J., Nakamoto, A., Negam, S., Ota, K., Chromatic Numbers of Quadrangulations on Closed Surfaces, J. Graph Theory 37 (2001), 100–114. [2] Thomassen, C., Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface, Trans. Amer. Math. Soc. 323 (1991), 605–635.

24 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

A characterization of a singular unicyclic graph

Milan Nath, Bhaba Kumar Sarma*

Department of Mathematics, IIT Guwahati, Guwahati 781039, Assam, India. [email protected]

2000 Mathematics Subject Classification. 05C05, 05C50, 15A18 A graph is said to be singular (nonsingular) if its adjacency matrix is sin- gular (nonsingular). The problem of characterizing a singular graph by its graph theoretic properties was first posed by Collatz and Sinogowitz [1] al- most fifty years back. The search for such a characterization is relevant in many disciplines of science which use graphs. For example, the occurrence of a zero eigenvalue in the spectrum of the adjacency matrix of the graph associated to the structure of a molecule (e.g. that of a hydrocarbon) indi- cates chemical instability of the molecule. Though several partial answers to this central question are known (see [2, 3, 4] for example), a characterization of a general singular graph by its graph properties is still elusive. In this short communication, we produce a (graph-theoretic) necessary condition for a graph to be singular, and show that the condition is also sufficient for trees and unicyclic graphs. Using this characterization, we de- termine the complete list of all nonsingular unicyclic graphs. Moreover, a sufficient condition for a general graph to be singular is supplied.

References

[1] Collatz L., Sinogowitz U., Spektren endlicher Grafen, Abh. Math. Sem. Univ. Hamburg 21 (1957), 63–77. [2] D. M. Cvetkovi´c, M. Doob, and H. Sachs, Spectra of Graphs. Academic Press, New York, 1980. [3] Fiorini S., Gutman I., Sciriha I., Trees with maximum nullity, Linear Algebra Appl. 397 (2005), 245–251. [4] Xuezhong T., Liu B., On the nullity of unicyclic graphs, Linear Algebra Appl. 408 (2005) 212–220.

ICM 2006 – Madrid, 22-30 August 2006 25 ICM 2006 – Short Communications. Abstracts. Section 14

The Norton algebra of a dual polar graph

Fernando Levstein*, Carolina Maldonado, Daniel Penazzi

FaMAF, Universidad Nacional de C´ordoba,Ciudad Universitaria (5000), C´ordoba, Argentina [email protected]

2000 Mathematics Subject Classification. 05E30 Let X be an association scheme of the type considered in [3]. We consider 2 2 P the operator L : L (X) 7→ L (X) given by L(f)(x) = y:d(x,y)=1 f(y). Then 2 L (Theorem 2.6 of [3]) L (X) = λ Vλ, where the Vλ ’s are the eigenspaces of L . Each Vλ carries a Norton algebra structure (see [2]) as follows: f ? g = 2 πλ(fg) where πλ : L (X) 7→ Vλ is the orthogonal projection and fg(x) = f(x)g(x). P. Terwilliger suggested us to consider the problem of finding a “nice” basis for (Vλ,?) in the case of dual polar graphs ([1]). He had done this for the Johnson scheme when λ is the second largest eigenvalue. First we can extend this to the Grassmann scheme. In those cases the product of two basis elements is a multiple of their sum. In the dual polar case we find a linearly generating set B for Vλ which is indexed by the corresponding polar graphs. Its elements behave like an orthogonal basis for projections P onto Vλ, (i.e., πλ(v) = c x∈B < v, x > x for some constant c) and their products are almost a multiple of the sum. We obtain a similar result for the Hamming scheme.

References

[1] Brouwer, A., Cohen, A.,Neumaier, Distance-Regular Graphs. Springer-Verlag, 1989. [2] Cameron, P., Goethals,J., Seidel, J., The Krein condition, spherical designs, Norton algebras and permutation groups, Proc. Kon. Nederl. Akad. Wetensch. (A) 81 (1978), 196–206. [3] Stanton, D., Orthogonal polynomials and Chevalley groups. In Special Func- tions:Group Theoretical Aspects and Applications (Eds Askey, Koornwinder, and Schempp); D. Reidel Publishing Company, (1984). 87–128.

26 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

Coloring squares of Kneser and Schrijver graphs

Jun-Yo Chen, Ko-Wei Lih*, and Jiaojiao Wu

Institute of Mathematics, Academia Sinica Sinica, Nankang, Taipei 115, Taiwan [email protected]

2000 Mathematics Subject Classification. 05C15 The Kneser graph KG(n, k) is the graph whose vertex set consists of all k- subsets of an n-set, and two vertices are adjacent if and only if they are disjoint. The Schrijver graph SG(n, k) is the subgraph of KG(n, k) induced by all vertices that are 2-stable subsets. The square G2 of a graph G is defined on the vertex set of G such that distinct vertices within distance two in G are joined by an edge. The span λ(G) of G is the smallest integer m such that an L(2, 1)-labeling of G can be constructed using labels belonging to the set 2 {0, 1, . . . , m}. The following results are established. (1) χ(KG (2k + 1, k)) 6 2 3k + 2 for k > 3 and χ(KG (9, 4)) 6 12. These improve results established in Kim and Nakprasit [1]. (2) χ(SG2(2k + 2, k)) = λ(SG(2k + 2, k)) = 2k + 2 2 2 for k > 4, χ(SG (8, 3)) = 8, λ(SG(8, 3)) = 9, χ(SG (6, 2)) = 9, and λ(SG(6, 2)) = 8.

References

[1] Kim, S.-J., Nakprasit, K., On the Chromatic Number of the Square of the Kneser Graph K(2k + 1, k), Graphs and Combin. 20 (2004), 70–90.

ICM 2006 – Madrid, 22-30 August 2006 27 ICM 2006 – Short Communications. Abstracts. Section 14

Measures of Vulnerability - the Tenacity Family

Dara Moazzami Faculty of Engineering, Department of Engineering Science, University of Tehran, P.O. Box:14395-195, Tehran IRAN [email protected]

2000 Mathematics Subject Classification. 05 The effective design of a survivable communications network requires a means of accurately evaluating its structural vulnerability both as a whole and with respect to its individual resources. For a communications network operating in a tactical environment, this evaluation should be based on a worst-case assumption that the enemy will selectively target those resources most critical to its topological integrity. A critical concern of overall system survivability, therefore, must be the specific level of connectivity associated with the topological structure of the supporting communications network. Since it is not clear which networks constitute “optimal networks”, the best we can do is to find some measure or measures that we believe do a reasonable job at measuring “goodness”. Many graph theoretical parameters have been used to describe the vul- nerability of communication networks, including toughness, binding num- ber, rate of disruption, neighbor-connectivity, integrity, mean integrity, edge- connectivity vector, l-connectivity and tenacity. In this paper we discuss tenacity and its properties in vulnerability calculation. We present several properties and bounds on the edge-tenacity. We also compute the edge- tenacity of some classes of graphs.

References

[1] Cozzens, M., D. Moazzami, and S. Stueckle, The tenacity of the Harary Graphs, J. Combin. Math. Combin. Comput. 16 (1994), 33-56. [2] Cozzens, M., D. Moazzami, and S. Stueckle, The tenacity of a Graph, Graph Theory, Combinatorics, and algorithms (Yousef Alavi and Allen Schwenk edds.) Wiley, New York, (1995), 1111-1112. [3] Moazzami, D., Vulnerability in Graphs - a Comparative Survey, J. Combin. Math. Combin. Comput. 30 (1999), 23-31. [4] Moazzami, D., Stability Measure of a Graph- a Survey, J. Utilitas Mathematica 57 (2000), 171-191. [5] Moazzami, D., On Networks with Maximum Graphical Structure and tenacity T, J. Combin. Math. Combin. Comput. 39 (2001) 121-126.

28 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

A special basis for the set-inclusion matrix Mtk(v)

Morteza Mohammad-Noori

Department of Mathematics, University of Tehran, Tehran, Iran [email protected]

2000 Mathematics Subject Classification. 05B05, 15A03

Given t, k, and v, the set-inclusion matrix Mtk(v) is defined as a (0, 1)- St matrix of inclusions for elements of i=0 Pi(v) versus elements of Pk(v): Mtk(v)(I,K) = 1 if I ⊆ K, and Mtk(v)(I,K) = 0 otherwise. It is known v
that if t ≤ k ≤ v−t, then rankZ (Mtk(v)) = t [1, 2]. Hence one may choose some of rows to form a basis for the row space of Mtk(v). We study one such special basis in details.

References

[1] De Caen, D., A note on the rank of set-inclusion matrices, The Electronic J. of Combin. 8 (2001). [2] Wilson, R. M., A diagonal form for the incidence Matrices of t-subsets vs. k-subsets, Europ. J. Combin. 11 (1990), 609–615.

ICM 2006 – Madrid, 22-30 August 2006 29 ICM 2006 – Short Communications. Abstracts. Section 14

Towards a unified theory of self-dual codes over Z2 × Z2

Bernardo R. Marquez

Department of Computer Science, Ateneo de Naga University, Ateneo Avenue, 4400 Naga City, Philippines [email protected]

2000 Mathematics Subject Classification. 94B Self-dual codes constitute an important class of codes primarily because of their deep algebraic and combinatorial properties. The nature of these codes depends on the alphabet and the inner product that are used to define them (see [4] for various ways of defining self-dual codes; in particular, see [1] and [3] for defining self-dual codes using the character group of a finite abelian group). This talk will present a unifying approach to the study of self-dual codes over the group Z2 × Z2. By defining a self-dual code with respect to a 2 x 2 binary matrix and also with respect to a set of such matrices, different fam- ilies of self-dual codes over Z2 × Z2 can be characterized by certain groups of 2 x 2 binary matrices. These families of codes include the Euclidean self- dual codes over the commutative rings on Z2 × Z2 and the self-dual codes over the field F4 with the Hermitian inner product, with the Euclidean trace inner product, and with the Hermitian trace inner product. The immediate goal is to formulate a complete characterization theorem for self-dual codes over Z2 × Z2 involving all the groups of 2 x 2 binary matrices. The ultimate goal is to come up with generalized versions of the concepts or results that appear in the traditional theory of self-dual codes over the finite fields or rings.

References

[1] Bannai, E., Modular Invariance Property of Association Schemes, Type II Codes over Finite Rings and Finite Abelian Groups, and Reminiscences of Francois Jaeger (A Survey), Ann. Inst. Fourier 49 (1999), 763–782. [2] Bannai, E., Munemasa, A., Duality Maps of Finite Abelian Groups and their Applications to Spin Models, J. Algebraic Combinatorics 8 (1998), 223–234. [3] Delsarte, P., An Algebraic Approach to the Association Schemes of Coding Theory. Philips Res. Repts. Suppl. 10, 1973. [4] Rains, E. M., Sloane, N. J. A., Self-dual Codes. In Handbook of Coding Theory (ed. by V. S. Pless and W. C. Huffman). Elsevier Science B. V., 1998, 177–294.

30 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

Supermodular functions on finite lattices

S. David Promislow*, Virginia R. Young

Department of Mathematics and Statistics, York University, Toronto, Canada, Department of Mathematics, University of Michigan, Ann Arbor, U.S.A

2000 Mathematics Subject Classification. 05, 06 A real valued function f on a lattice L is said to be supermodular if f(x ∨ y) + f(x ∧ y) ≥ f(x) + f(y), for all elements x, y ∈ L. If equality holds, f is said to be modular. We consider finite lattices and the problem of determining the extreme rays of the cone S/M, in which S is the cone of supermodular functions and M is the vector space of modular functions. This problem arises in different contexts. In probability theory, it is of interest in connection with the supermodular order on random vectors, a popular method of comparing these according to their degree of riski- ness and variability. It also has connections with polymatroid theory and is equivalent to a problem raised by Jack Edmonds in 1990, for which little progress has been made up the present time. We are able to solve the problem completely for the simplest type of lattices, namely those which are the disjoint union of chains, with the 0 k and 1 elements identified. We investigate the lattice, ZN consisting of all k-tuples with entries from the set {0, 1,...,N −1}, equipped with the usual 2 3 pointwise order. We provide complete answers for the cases of ZN , Z2 , 4 3 Z2 , and Z3 , and we have a conjecture for the general case.

ICM 2006 – Madrid, 22-30 August 2006 31 ICM 2006 – Short Communications. Abstracts. Section 14

How far can we go with edge transitivity in hypergraphs?

S`oniaP. Mansilla*, Oriol Serra Department of Applied Mathematics IV, Technical University of Catalonia, Avda. del Canal Ol´ımpic, s/n, 08860 Castelldefels, Spain [email protected]

2000 Mathematics Subject Classification. 05C65, 05C25, 05B05, 51E05 A hypergraph is s-edge transitive if its automorphism group acts transi- tively on the set of its s-edges. In this talk, we will study s-edge transitive hypergraphs and prove that if a hypergraph has degree at least three and all edges of size at least two, then s ≤ 5. Besides, given an s-edge transitive hypergraph, we can prove that there are infinitely many s-edge transitive hypergraphs that cover the given one. Bounds of s-edge transitivity in graphs have been the subject of much investigation. The study of s-edge transitive graphs goes back to Tutte [3], who showed that finite cubic graphs cannot be s-edge transitive for s > 5. Weiss [4] proved several years later that the only finite connected s-edge transitive graphs with s ≥ 8 are the cycles. Given G ≤ AutΓ, we say that Γ is locally (G, s)-edge transitive if for each vertex v, the stabiliser Gv acts transitively on the set of s-edges starting at v. Provided that all vertices have degree at least three, a locally s-edge transitive graph is also locally (s − 1)-edge transitive. If G is not vertex transitive, then a locally 2-edge transitive graph is bipartite and the two parts of the bipartition are G-orbits. Stellmacher [2] has proved that if G is vertex intransitive and all vertices have degree at least three, then s ≤ 9. This inequality is sharp as demonstrated by the incidence graphs of the 2 classical generalised octagons associated with the simple groups F4(q). The situation is different in the directed case. Mansilla and Serra [1] showed that given an arbitrary regular digraph and an arbitrary positive integer s, there are infinitely many s-arc transitive digraphs which cover the original digraph, so in particular there are infinite families of s-arc transitive digraphs for each positive integer s and each degree. Following previous work by the authors, in this talk the definitions of s-edge transitivity are extended to the case of hypergraphs and it will be shown that the bound for s that applies in this case is 5.

References

[1] S. P. Mansilla, O. Serra, Construction of k-arc transitive digraphs, Discrete Math. 231 (2001) 337–349.

32 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

[2] B. Stellmacher, Locally s-transitive graphs, unpublished. [3] W. T. Tutte, On the symmetry of cubic graphs, C anad. J. Math. 11 (1959) 621–624. [4] R. Weiss, The nonexistence of 8-transitive graphs, C ombinatorica 1 (1981) 309–311.

ICM 2006 – Madrid, 22-30 August 2006 33 ICM 2006 – Short Communications. Abstracts. Section 14 k-fixed-points-permutations

Fanja Rakotondrajao

D´epartement de Math´ematiques et Informatique, Universit´ed’Antananarivo, 101 Antananarivo, Madagascar [email protected]

2000 Mathematics Subject Classification. 05A15, 05A19 k We study in this paper the new numbers dn which enumerate the k-fixed- points-permutations, that is, permutations σ such that for all integers i in the interval [k], σp(i) ∈/ [k] \{i} for all nonnegative integer p and the set of fixed points of σ F ix(σ) ⊆ [k]. The first few values of these numbers are given in the following table

k dn k = 0 1 2 3 4 5 n = 0 1 1 0 1 2 1 1 1 3 2 3 2 1 4 9 11 7 3 1 5 44 53 32 13 4 1

We will give a combinatorial interpretation to each of the following relations defining them

k dk =1 k k k dn =(n − 1)dn−1 + (n − k − 1)dn−2, for n ≥ k ≥ 0 k k−1 k−1 ndn =dn−1 + dn , for n ≥ k ≥ 0 k k−1 k dn + dn−2 = ndn−1, for n ≥ k ≥ 1.

A combinatorial interpretation directly on derangements for k = 0 of the last relation, which is the famous relation of the derangement numbers n−1 dn + (−1) = ndn−1 ([1], [2], [3]), is given. The generating functions of k the numbers dn have the closed form

un exp(−u) D(k)(u) = P dk = , n≥0 n+k n! (1 − u)k+1 un exp(−u) D(x, u)= P P dk xk = . k≥0 n≥0 n+k n! 1 − x − u

34 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

References

[1] Desarmenien, J., Une autre interpr´etationdu nombre de d´erangements, In Actes 8e S´em.Lothar. Comb., IRMA Strasbourg 1984, 11–16. [2] Mantaci,R., Rakotondrajao,F., Exceedingly deranging!, Advances in Applied Mathematics, 30 Issue 1 / 2 (January 2003), 177–188. [3] Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, New York, 1958.

ICM 2006 – Madrid, 22-30 August 2006 35 ICM 2006 – Short Communications. Abstracts. Section 14

The weight distributions of irreducible cyclic codes of length 2m

G. K. Bakshi, M. Raka, Anuradha Sharma*

Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, India aradha−[email protected]

2000 Mathematics Subject Classification. 05A15 Let m be a positive integer and q be an odd prime power. In this short communication, the weight distributions of all the irreducible cyclic codes m of length 2 over Fq are determined explicitly, directly from their generating polynomials.

References

[1] Baumert, L. D., McEliece, R. J., Weights of irreducible cyclic codes, Informa- tion and Control 20 (1972), 158–175. [2] Fitzgerald, R. W., Yucas, J. L., Sums of Gauss sums and weights of irreducible codes, Finite Fields Appl. 11 (2005), no. 1, 89–110. [3] MacWilliams, F. J., Seery, J., The weight distributions of some minimal cyclic codes, IEEE Trans. Inform. Theory 27(1981), no. 6, 796-806. [4] Moisio, M. J., Va ¨a¨n¨anen, K. O., Two recursive algorithms for computing the weight distribution of certain irreducible cyclic codes, IEEE Trans. Inform. Theory 45 (1999), no. 4, 1244–1249. [5] Van Der Vlugt, Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes, J. Number Theory 55 (1995), no. 2, 145–159.

36 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

Generalized splitting operation for binary matroids and its applications

M. M. Shikare Department of Mathematics, University of Pune, Pune-411007, India [email protected]

2000 Mathematics Subject Classification. 05B35 We define a generalized splitting operation for a binary matroid in the following way: Let M be a binary matroid on a set S and T be a subset of S. Suppose that A is a matrix over GF(2) that represents the matroid M. Let AT be the matrix that is obtained by adjoining an extra row to A with this row being zero everywhere except in the columns corresponding to the elements of T where they take the value 1. Let MT be the matroid represented by the matrix AT . We say that MT has been obtained from M by splitting the set T. The transition from M to MT is called a generalized splitting operation. We prove the following Theorems. Theorem 1: Let M be a binary matroid on a set S having the set of circuits C and let T be a subset of S. Then the circuits of MT consists of minimal members of the set C0 = {C0 ⊆ S : |C0 ∩ T | ≡ 0(mod 2) and C0 is either a circuit of M or a union of two disjoint circuits of M}. Theorem 2: Let M be a binary matroid on a set S, and T be a subset of S. Suppose that M has a circuit that contains an odd number of elements of T . A subset BT of S is a basis of MT if and only if BT = B ∪ {α} where B is a basis of M, α ∈ S − B, and the unique circuit of M contained in BT contains an odd number of elements of T . We say that a binary matroid N on the set S is an identification of M with respect to a subset T of S if NT = M. Theorem 3: Let M be a rank-k binary matroid on a set S, and T be the support of a row of A. Then M has precisely 2k−1 +1 distinct identifications with respect to T . Conditions for the splitting operation to preserve connectedness of a binary matroid are given. We use this operation to characterize binary Euler matoids and affinely independent column vectors of a binary matrix. Some other applications of this operation have been given.

References

[1] J. G. Oxley, Matroid Theory, Oxford University Press, Oxford (1992).

ICM 2006 – Madrid, 22-30 August 2006 37 ICM 2006 – Short Communications. Abstracts. Section 14

[2] T. T. Raghunathan, M. M. Shikare and B. N. Waphare, Splitting in a binary matroid, Discrete Math. 184 (1998), 267-271. [3] M. M. Shikare and G. Azadi, Determination of the bases of a splitting matroid, European J. of combinatorics 24 (2003), 45-52. [4] P. J. Slater, A classification of 4-connected graphs, J. Combin. Theory 17 (1974), 282-298. [5] D. J. A. Welsh, Euler and bipartite matroids, J. Combin. Theory 6 (1969), 375-377.

38 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

Defensive alliances in graphs

J. A. Rodr´ıguez, J. M. Sigarreta Department of Computer Engineering and Mathematics, Rovira i Virgili University of Tarragona, Av. Pa¨ısos Catalans 26, 43007 Tarragona, Spain; Department of Mathematics, University Carlos III of Madrid, Avda. de la Universidad 30, 28911 Legan´es(Madrid), Spain [email protected]; [email protected]

2000 Mathematics Subject Classification. 05C69; 15C05 A defensive alliance in a graph Γ = (V,E) is a set of vertices S ⊂ V where for every vertex v ∈ S, at least half of the vertices in the closed neighborhood of v are in S. An alliance S is called global if it affects every vertex in V \S, that is, S is a dominating set of Γ. The defensive alliance number a(Γ) is the minimum cardinality of a defensive alliance in Γ. The global defensive alliance number γa(Γ) is defined similarly. Clearly, a(Γ) ≤ γa(Γ). The study of defensive alliances in graphs, together with a variety of other kinds of alliances, was introduced in [2]. In particular, several bounds on the defensive alliance number were given. For instance, it was shown n that a(Γ) ≤ 2 , where n denotes de order of Γ. Moreover, the graphs having a(Γ) ≤ 3 were characterized in the referred paper. The particular case of global defensive alliance was investigated in [1] where was shown that if Γ n has maximum degree ∆, γa(Γ) ≥ ∆ . d 2 e+1 We obtain several tight non-trivial bounds on a(Γ) and γa(Γ) in terms l nµ m of several parameters of Γ. For instance, we show that a(Γ) ≥ n+µ and l n m γa(Γ) ≥ λ+2 , where µ denotes the algebraic connectivity of Γ and λ denotes its spectral radius. The case of strong alliances is studied by analogy. Moreover, we study the particular case of alliances in planar graphs. For instance, we show that if Γ is a planar graph of order n > 6, then γa(Γ) ≥ l n+12 m l n+8 m 8 and if n > 6 and Γ is a triangle-free graph, then γa(Γ) ≥ 6 . In the case of the line graph, L(Γ), of a simple graph Γ, we show that δ + δ − 1 l√ m n n−1 ≤ a(L(Γ)) ≤ δ and γ (L(Γ)) ≥ m + 4 − 1 . 2 1 a where δ1 ≥ · · · ≥ δn is the degree sequence of Γ and m > 6 denotes its size. In particular, if Γ is a δ-regular graph (δ > 0), then a(L(Γ)) = δ, and if l m δ1+δ2−1 Γ is a (δ1, δ2)-semiregular bipartite graph, then a(L(Γ)) = 2 . As a consequence of the study we compare a(L(Γ)) and a(Γ), and we characterize the graphs having a(L(Γ)) ≤ 3. The case of strong alliances is studied by analogy.

ICM 2006 – Madrid, 22-30 August 2006 39 ICM 2006 – Short Communications. Abstracts. Section 14

References

[1] T. W. Haynes, S. T. Hedetniemi, and M. A. Henning, Global defensive alliances in graphs, Electron. J. Combin. 10 (2003), Research paper 47. [2] P. Kristiansen, S. M. Hedetniemi and S. T. Hedetniemi, Alliances in graphs. J. Combin. Math. Combin. Comput. 48 (2004), 157-177.

40 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

On the discrete time, cost and quality trade-off problem

Hassan Taheri, Hamed R. Tareghian*

Department of Mathematics, University of Khayam, Mashhad, Iran; Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran [email protected]

2000 Mathematics Subject Classification. 90B50 n this paper a solution procedure is developed to study the tradeoffs among time, cost and quality in the management of a project. This problem as- sumes the duration and quality of project activities to be discrete, non- increasing functions of a single nonrenewable resource. Three inter-related integer programming models are developed such that each model optimizes one of the given entities by assigning desired bounds on the other two. Different forms of quality aggregations and effect of activity mode reduc- tions are also investigated. The computational performance of the models is presented using a numerical example.

References

[1] De, P., Dunne, E.J., Gosh, J.B., Wells, C.E. The discrete time-cost tradeoff problem revisited. European Journal of Operational 81 (1995) 225-238. [2] Demeulemeester, E., Herroelen, W., Elmaghraby, S.E. Optimal procedures for the discrete time/cost trade-off problem in project networks. European Journal of Operational Research 88 (1996) 50-68. [3] Deineko, V.G., Woeginger, G.J., Hardness of approximation of the discrete time-cost tradeoff problem. Operations Research Letters 29 (2001) 207-210. [4] Khang, D. B., Myint, Y. M. Time, cost and quality trade-off in project man- agement: a case study. International Journal of Project Management Vol. 17, No. 4 (1999) 249-256. [5] Agrawal, M.K., Elmaghraby, S.E., Herroelen, W. DAGEN: A generator of test sets for project activity nets. European Journal of Operations Research, 90 (1996) 376-382.

ICM 2006 – Madrid, 22-30 August 2006 41 ICM 2006 – Short Communications. Abstracts. Section 14

Characterizations of the finite split Cayley hexagons by intersection numbers

J. A. Thas* and H. Van Maldeghem

Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, 9000 Gent, Belgium [email protected]

2000 Mathematics Subject Classification. 51E12 The following characterizations of the finite split Cayley hexagons [2] by intersection numbers are obtained. A set L of lines of the finite projective space PG(5, q), q 6= 2, 3, with the properties that (1) every plane contains 0,1 or q + 1 elements of L, (2) every 3-dimensional subspace contains no more than q2 + q + 1 and no less than q + 1 elements of L, (3) every point of PG(5, q) is on q + 1 members of L, and (4) there is at least one point x for which all lines of L on x are contained in a hyperplane, is necessarily the set of lines of a regularly embedded [1] split Cayley hexagon H(q). For q = 2, 3 we have the same conclusion if (4) is replaced by: there is at least one point x of PG(5, q) such that all lines of L on x are in a plane, 3-space, respectively. Next, let L be a set of lines of PG(6, q) with the properties that (1) every point of PG(6, q) on at least one line of L is on exactly q + 1 elements of L, (2) every plane of PG(6, q) contains either 0,1 or q + 1 elements of L, (3) every 3-space of PG(6, q) contains either 0, 1, q + 1 or 2q + 1 elements of L, (4) every hyperplane of PG(6, q) contains at most q3 + 3q2 + 3q lines of L, and (5) L contains q5 + q4 + q3 + q2 + q + 1 lines. Then L is the line set of a regularly embedded split Cayley hexagon H(q) in PG(6, q).

References

[1] Thas, J. A., Van Maldeghem, H., Flat lax and weak lax embeddings of finite generalized hexagons, European J. Combin., 19 (1998), 733–751. [2] Van Maldeghem, H., Generalized Polygons, Monographs in Mathematics 93, Birkh¨auserVerlag, Basel, 1998.

42 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

On self duality of pathwidth in polyhedral graph embeddings Fedor V. Fomin, Dimitrios M. Thilikos* Department of Informatics, University of Bergen, N-5020 Bergen, Norway; Departament de Llenguatges i Sistemes Inform`atics, Universitat Polit`ecnica de Catalunya, 08034 Barcelona, Spain [email protected]; [email protected] 2000 Mathematics Subject Classification. 05C10, 05C83 For c ≥ 0, we say that a graph parameter p is additively c-self dual on a subclass G of plane graphs if for every graph G ∈ G and for its geomet- rical dual G∗, p(G∗) ≤ p(G) + c. In [1] Golovach conjectured that: For every 2-connected plane graph G the edge search number of G is equal to the node search number of its dual G∗. This Conjecture would imply that the pathwidth is additively 1-self dual on 2-connected plane graphs. There are two width parameters, related to pathwidth, namely, branchwidth and treewidth, known to be additively self dual. In case of branchwidth, it fol- lows from the results in [3] that it is additively 1-self dual for planar graphs in general and additively 0-self dual for planar graphs that are not trees. The additively 1-self duality of treewidth was first proved in [2]. In this pa- per we we disprove the above conjecture by proving that for each number c ≥ 0 pathwidth is not additively c-self dual on 2-connected plane graphs. In order to further explore the self duality of pathwidth, we define a weaker version of it: A graph parameter p is multiplicatively c-self dual for a sub- class G of plane graphs if there exists a constant d ≥ 0 such that for every graph G ∈ G and its dual G∗, p(G∗) ≤ c · p(G) + d. We prove that, for simple 3-connected planar graphs, pathwidth is multiplicatively 6-self dual. Actually we prove a more general result: for every polyhedral embedding of a graph G in some surface of oriented genus g, the pathwidth of G is at most 6 · (pathwidth(G∗) + g − 2), where G∗ is the geometric dual of G. On the other side, we show that on 3-connected planar graphs, pathwidth fails to be multiplicatively c-self dual for every c < 1.5.

References

[1] Petr A. Golovach. Extremal Search Problems on Graphs. PhD thesis. Leningrad State University, 1990. [2] D. Lapoire. Treewidth and duality in planar hypergraphs. See web page: http://www.labri.fr/Perso/ ˜lapoire/papers/dual planar treewidth.ps [3] P. D. Seymour and R. Thomas, Call routing and the ratcatcher, Combina- torica, 14 (1994), 217–241.

ICM 2006 – Madrid, 22-30 August 2006 43 ICM 2006 – Short Communications. Abstracts. Section 14

Combinatorics related to continued fractions

Flemming Topsøe

Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark [email protected]

2000 Mathematics Subject Classification. 11Y65, 11B39, 68R05 A family over an interval [i, j] consists of n = j − i + 1 members, one for each “age category”. Each member is either a mother, a daughter or a free woman. The mothers have only one daughter each. The daughters are one age class younger than their respective mothers. Thus, the youngest family member cannot be a mother. With these rules, we see that the population j P n−ν
Ωi of all families over [i, j] consists of ν ν = Fn+1 families (with F denoting Fibonacci numbers). For the connection to continued fractions consider e.g. the 8.th approx- imant

a1 a2 a3 a4 a5 a6 a7 a8 A8 b0 + = b1+ b2+ b3+ b4+ b5+ b6+ b7+ b8 B8 of a continued fraction. Then A8 and B8 are certain multi-polynomials. One of the terms in A8 is a1a4a7b2b5b8 and this term corresponds to the family DMFDMFDMF over [0, 8] which is characterized by the three mothers of ages 1, 4 and 7. Every mother gives rise to an a-factor in the term of A8, every free woman to a b-factor, whereas the daughters do not contribute to the term. Thus, a1a4a7b2b5b8 is the contribution to A8 from the family 8 DMFDMFDMF. Adding all contributions from families in Ω0, the numera- tor A8 is obtained. Similarly, B8 is the sum of contributions from all families 8 in Ω1. Generalizing this, we see that the approximants of a continued fraction can be obtained by combining purely combinatorial constructs with ana- lytic elements. Though hardly more expedient than standard approaches via recurrence relations, the combinatorial view provides new insight and offers new opportunities. For instance, the important determinant identities can be derived by a simple combinatorial argument. j Central to our study is the consideration of the partition functions Zi , defined as the total contribution, in the spirit of the above examples, from j families in Ωi . The combinatorial approach suggests certain probabilistic models in case the a’s and b’s are positive. Limit models for infinitely large families

44 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14 can be constructed if and only if the continued fraction is convergent. In this case, the demographic factor can be defined. It represents the relative number of mothers in very large families. To summarize: Simple constructs of a combinatorial nature are sug- gested which throws new light on the basics of continued fraction theory. Though related to a classical result, the Euler-Minding formula, it appears that the possibilities for extra insight of a combinatorial nature have been overlooked during the long development since the times of the old masters.

ICM 2006 – Madrid, 22-30 August 2006 45 ICM 2006 – Short Communications. Abstracts. Section 14

Log-concavity and LC-positivity

Yi Wang, Yeong-Nan Yeh*

Institute of Math.,Academia Sinica,Taipei 11529, Taiwan [email protected]

2000 Mathematics Subject Classification. 05A20; 15A04; 05A15; 15A48

A triangle {a(n, k)}0≤k≤n of nonnegative numbers is LC-positive if for Pn k each r, the sequence of polynomials k=r a(n, k)q is q-log-concave. It is double LC-positive if both triangles {a(n, k)} and {a(n, n − k)} are LC- positive. We show that if {a(n, k)} is LC-positive then the log-concavity of the sequence {xk} implies that of the sequence {zn} defined by zn = Pn k=0 a(n, k)xk, and if {a(n, k)} is double LC-positive then the log-concavity of sequences {xk} and {yk} implies that of the sequence {zn} defined by Pn zn = k=0 a(n, k)xkyn−k. Examples of double LC-positive triangles include the constant triangle and the Pascal triangle. We also give a generalization of a result of Liggett that is used to prove a conjecture of Pemantle on characteristics of negative dependence.

References

[1] F. Brenti, Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, Contemp. Math. 178 (1994) 71-89. [2] T. M. Liggett, Ultra logconcave sequence and negative dependence, J. Combin. Theory Ser. A 79 (1997) 315-325. [3] B. E. Sagan, Log concave sequences of symmetric functions and analogs of the Jacobi-Trudi determinants, Trans. Amer. Math. Soc. 329 (1992) 795–811. [4] R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Ann. New York Acad. Sci. 576 (1989) 500–534. [5] Y. Wang and Y.-N. Yeh, Polynomials with real zeros and P´olya frequency sequences, J. Combin. Theory Ser. A 109 (2005) 63–74.

46 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

Non-archimedean polytopes and cellular resolutions

Josephine Yu

Department of Mathematics, University of California, Berkeley, CA 94720, USA [email protected]

2000 Mathematics Subject Classification. 52B99 Tropical convex hulls of finite sets of points are analogues of usual convex hulls, in the geometry over the tropical semiring (R, max,+). In this talk, we will look at two connections between these and cellular resolutions. Tropical polytopes are usual polyhedral complexes and have natural structures of cellular free resolutions, i.e., their combinatorial data give rise to minimal free resolutions for some monomial ideals. On the other hand, they are also images of usual polytopes in an affine space over the Puiseux series field, under a non-archimedean valuation into the real numbers. For any monomial ideal, there is a family of polytopes over the Puiseux series field that gives cellular resolutions of that monomial ideal. This family includes the hull complexes.

References

[1] Florian Block and Josephine Yu, Tropical Convexity via Cellular Resolutions, Preprint: math/0503279. To appear in the Journal of Algebraic Combinatorics. [2] Mike Develin and Josephine Yu, Non-Archimedean Polytopes and Cellular Res- olutions, in preparation.

ICM 2006 – Madrid, 22-30 August 2006 47 ICM 2006 – Short Communications. Abstracts. Section 14

Some relations between rank and energy of graphs

Sanaz Zare

Department of Mathematical Sciences, Sharif University of Technology, Azadi Street, P.O. Box 11365-9415, Tehran, Iran sa zare [email protected]

2000 Mathematics Subject Classification. 05C50, 15A03, 15A18 Let G be a graph and the rank of its adjacency matrix denoted by rank(G), see [1]. The energy of a graph is defined as the sum of the absolute values of all eigenvalues of a graph and denoted by E(G), see [2], [3], [4]. In this paper we characterize all graphs whose E(G) = rank(G). Let G be a graph of order n. We prove that E(G) ≥ rank(G) and the equality holds if and r only if G = 2 K2 ∪(n−r)K1, for some positive integer r. For every connected bipartite graph G of rank r, it is shown that E(G) ≥ p(r + 1)2 − 5. We prove that if T is a singular tree with at least 4 vertices, then E(T ) ≥ 1 + rank(T ). A graph G of order n is called hyperenergetic if E(G) > 2n − 2, where E(G) is the energy of G. In this paper we prove that the Kneser graph Kn:r is hyperenergetic for any natural numbers n and r ≥ 2 with n ≥ 2r + 1. Also we prove that for r ≥ 2, the complement of Kneser graph, E(Kn:r), is hyperenergetic.

References

[1] Akbari, S., Cameron, P. G., Khosrovshahi, G. B., Ranks and signatures of adjacency matrices, sumitted to J. Graph Theory. [2] Balakrishnan, R. The energy of a graph, Linear Algebra Appl. 387(2004), 287- 295. [3] Gutman, I. The energy of graph: old and new results , in: A. Betten, A.Kohnert, R. Laue. A. Wassermann(Eds.), Algebraic Combinatorics and Applications, Springer-Verlag, Berlin, 2001, 196-211. [4] Fajtlowicz, S. On the conjectures of Graffit II, Cong. Num. 60 (1987), 189-197.

48 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14

On the Vertex Distinguishing Equitable Total Chromatic Number of Graphs

Zhang Zhongfu

Institute of Applied Mathematics, Lanzhou Jiaotong University,730070,P.R.China zhang zhong [email protected]

2000 Mathematics Subject Classification. 05C15 In this paper, we present a new concept which vertex distinguishing eq- uitable total coloring of a graph (Let G be a connected graph with or- der at least 2, k is a positive integer and f is the mapping from V (G) ∪ E(G) to {1, 2, . . . , k}. For any v ∈ V (G). If (1) ∀uv, vw ∈ E(G), u 6= w, we have f(uv) 6= f(vw); (2) ∀uv ∈ E(G), u 6= v, we have f(u) 6= f(v), f(u) 6= f(uv), f(v) 6= f(uv); (3) ∀u, v ∈ V (G), u 6= v, we have C(u) 6= C(v), where C(u) = {f(u)} ∪ {f(uv)|uv ∈ E(G)}, then f is called a k-vertex distinguishing total coloring of graph G(in brief, k-VDTC) and the number χvt(G) = min{k|G has k-VDTC} is called the ver- tex distinguishing total chromatic number of graph G. If it also satis- fies (4)For i ∈ {1, 2, . . . , k}, denote {u ∈ V (G)|f(u) = i} and {uv ∈ E(G)|f(uv) = i} by Vi and Ei respectively and denote Vi ∪ Ei by Si. If for any i, j ∈ {1, 2, . . . , k}, we have ||Si| − |Sj|| ≤ 1, then f is called a k vertex distinguishing equitable total coloring of G( in brief k-VDETC).And the number χvet(G) = min{k|G has a k-VDETC of G} is called vertex dis- tinguishing equitable total chromatic number of G.), meanwhile, we have obtained the vertex distinguishing equitable total chromatic numbers of some graphs, such as complete graph,star,complete bipartite,complete bal- anced bipartite,wheel.fan,double star,and join graphs of path and path,path and cycle,cycle and cycle,path and star, and present two conjecture:(1) For any simple graph G, kT (G) ≤ χvet(G) ≤ kT (G) + 1,where kT (G) = k min{k|(d+1) ≥ nd}|δ(G) ≤ d ≤ ∆(G)} and nd is the number of vertices of G with degree d, (2)For any simple graph G, χvet(G) = χvt(G),where χvt(G) is vertex distinguishing total chromatic number.

References

[1] Balister,P.N., Bollob´as, B., Shelp,R.H., Vertex distinguishing colorings of graphs with ∆(G) = 2, Disc. Math. 252(2002), 17 ∼ 29. [2] Balister,P. N., Riordan,O.M. and Sclelp,R. H., Vertex distinguishing Coloring of Graphs,J.Graph Theo.,2003, 42:95 ∼ 109.

ICM 2006 – Madrid, 22-30 August 2006 49 ICM 2006 – Short Communications. Abstracts. Section 14

[3] Bazgan,C., Harkat-Benhamdine,A.,Li,H., Wo´zniak,M., On the vertex dis- tinguishing proper edge coloring of graphs,J. Combin. Theory Ser. B 75(1999)288 ∼ 301. [4] Burris,A.C., and Schelp,R.H., Vertex distinguishing proper edge colorings.J. Graph Theo., 1997, 26:73 ∼ 82. [5] Bondy,J.A. and Murty,U.S.R., Graph Theory with Applications, T he Macmil- lan press Ltd, 1976.

50 ICM 2006 – Madrid, 22-30 August 2006