DOCSLIB.ORG

Graphs with Large Girth and Large Chromatic Number

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Graphs with Large Girth and Large Chromatic Number GRAPHS WITH LARGE GIRTH AND LARGE CHROMATIC NUMBER CHEUK TO TSUI Abstract. This paper investigates graphs that have large girth and large chromatic number. We first give a construction of a family of graphs that do not contain cycles of length 3, but have unbounded chromatic number. Next, we prove the existence of graphs that have large girth and large chro- matic number. With existence proven, we give an explicit construction of such graphs. Contents 1. Introduction 1 2. Triangle-Free Graphs with Large Chromatic Number 2 3. Existence of Graphs with Large Girth and Large Chromatic Number 3 4. Construction of Graphs with Large Girth and Large Chromatic Number 5 4.1. Lower Bound on Girth of Xp 8 4.2. Lower Bound on Chromatic Number of Xp,q 11 Acknowledgments 13 References 14 1. Introduction Finding a lower bound for the chromatic number of a given graph is, in general, difficult to do. There are few techniques that show that a certain number of colors are not enough to color a graph. Because of this, it is interesting to consider how we may construct graphs with large chromatic number. One way to do so is to include many cycles; since every even cycle requires at least 2 colors and every odd cycle requires at least 3, it is easy to construct graphs with large chromatic number as long as cycles are permitted (take the complete graphs, for instance). But the matter becomes more complicated when we do not permit cycles below a given length, and want to construct graphs with large girth and large chromatic number. In the remainder of Section 1, we define the terminology that allows us to begin addressing this problem. In section 2, we address the simplified problem of showing that there are triangle-free graphs with arbitrarily large chromatic number. In section 3, we show more generally that there exist graphs that have girth l and chromatic number k for any k and l. In section 4, we explicitly construct≥ graphs with large girth and≥ chromatic number, show the basic method of finding a lower bound for their girth, and find a lower bound for their chromatic number. 1 2CHEUKTOTSUI Definition 1.1. A graph G =(V,E) consists of a vertex set V and an edge set E, where elements of the edge set E are unordered pairs vi,vj ,withvi = vj and v ,v V . { } 6 i j 2 Definition 1.2. Two vertices vi and vj are adjacent if there is an edge between them, that is, v ,v E. We can also write v v . { i j} 2 i ⇠ j Definition 1.3. A cycle is a sequence of vertices that begins and ends at the same vertex, such that successive vertices are adjacent and no vertex except the first repeats. The number of distinct vertices in the sequence is called the length of the cycle. A cycle of length k is denoted Ck. Definition 1.4. The girth of a graph is the length of the shortest cycle contained in it. If a graph contains no cycles, its girth is defined to be . 1 Definition 1.5. A graph G is triangle-free if it does not contain a cycle of length 3. Definition 1.6. A set of vertices S is independent if no two vertices in S are adjacent. Definition 1.7. The independence number ↵(G) of a graph G is the maximum size of an independent vertex set. Definition 1.8. A (proper) k-coloring of a graph is a function f : V 1, 2,...,k such that if v and v are adjacent, then f(v ) = f(v ). ! { } i j i 6 j Definition 1.9. The chromatic number χ(G) of a graph G is the minimum value of k such that a k-coloring exists. 2. Triangle-Free Graphs with Large Chromatic Number Before addressing the general question of constructing graphs with large girth and chromatic number, we first consider the simpler problem of constructing triangle- free graphs with large chromatic number. Theorem 2.1. For any positive integer k, there exists a triangle-free graph with chromatic number k. Proof. For k = 1, a single point will work. For k = 2, two adjacent vertices will work, and for k = 3, C5 will work. For chromatic numbers larger than 3, we give a construction that, when applied to a triangle-free graph, produces a new triangle-free graph that increases the chromatic number by 1. Construction 2.2 (Mycielski. See [1]). Given a graph G, label its vertices v ,i i 2 1, 2,...,n . Create n new vertices ui and an additional vertex w, and draw an edge between{ each} u and w.Then,if v ,v is an edge, draw the two edges u ,v i { i j} { i j} and v ,u . Call this new graph G0. { i j} First, we show that if G is triangle-free, then G0 is triangle-free. Suppose there exists a C3 in G0.Vertexw cannot be a vertex of a C3, because it is adjacent only to all the ui, which are not pairwise adjacent. There are also no C3’s containing two of the ui,sincenotwoui are adjacent. Therefore, the C3 must contain two adjacent vi and one of the ui. Let the vertices be vj, vk and ul. They must be pairwise adjacent, meaning that vj,ul and vk,ul are edges of G0. By the construction, this would be the case only{ if }v ,v {and }v ,v were edges of G. But this would { j l} { k l} GRAPHS WITH LARGE GIRTH AND LARGE CHROMATIC NUMBER 3 mean that vj,vk,vl,vj would be a C3 in G, which contradicts the fact that G is triangle-free. Thus there exists no C3 in G0. Now, we show that χ(G0)=χ(G) + 1. Let χ(G)=k. This means that the vi must be colored by at least k colors. We claim that in any coloring of G0,theui must also be colored by at least k colors. Suppose that the ui can be colored by at most k 1 colors. so at least one of the v are colored by color k, while none − i of the ui are. Observe that for all i, vi and ui are adjacent to exactly the same vj’s in G. Take the vi that are colored by color k, and change them to the color of the corresponding ui. This new coloring of the vi would still be valid, because each vi has the same neighbours as the corresponding ui, meaning that none of the vj’s adjacent to each vi have the same color as the corresponding ui.Butthis new coloring (of the v ) uses at most k 1 colors, contradicting the fact that G i − must be colored by at least k colors. Thus, the ui must be colored by at least k colors as well. Since w is adjacent to all the ui, it must be colored by a (k + 1)st color. Therefore, χ(G0) χ(G) + 1. Furthermore, G0 can be colored by χ(G)+1 ≥ colors: First color the vi with k colors, then color each ui the same color as the corresponding vi. Then color w with a (k + 1)st color. So χ(G0)=χ(G) + 1. Therefore, to obtain a triangle-free graph with chromatic number k>3, we iteratively apply the construction to C k 3 times. 5 − ⇤ 3. Existence of Graphs with Large Girth and Large Chromatic Number Theorem 2.1 shows that the chromatic number of a graph can be made arbitrarily large, even if it does not contain cycles of length 3. Now we consider the more general question of whether it is possible to avoid cycles below a given length, yet still have an arbitrarily large chromatic number. As preparation, we present one probabilistic result and define some more notation. Lemma 3.1 (Markov’s Inequality). If X is a non-negative discrete random variable and a>0, then E[X] P [X a] . ≥ a Proof. We have E[X]= xP (x) x 0 X≥ = xP (x)+ xP (x) x a x<a X≥ X xP (x) ≥ x a X≥ a P (x) ≥ x a X≥ = aP [X a]. ≥ ⇤ 4CHEUKTOTSUI Definition 3.2 ([2, p. 38]). We write G G(n, p)ifG is a random graph on n ver- tices chosen by picking each pair of vertices⇠ as an edge randomly and independently with probability p. Definition 3.3 (little-oh notation). We write an = o(bn)if a lim n =0. n !1 bn Remark 3.4. an = o(1) means that limn an =0. !1 Theorem 3.5 (Erd¨os1959. See [2, pp. 38-39]). For all k, l there exists a graph G with girth(G) >land χ(G) >k. ✓ 1 Proof. Let ✓ < 1/l, and let G G(n, p)withp = n − . Let X be the number of ⇠ cycles of length at most l in G, and let Xi be the number of cycles of size exactly i. By linearity of expectation, we have l E[X]= E[Xi]. i=3 X To determine E[X], we count the possible number of cycles of length i. There are n (i 1)! i ways to choose the vertices of the cycle, and −2 ways to order them (we can start with any vertex). We must divide by 2, because the reverse of any ordering gives the same cycle. So the total number of possible cycles is n (i 1)! (n) − = i , i 2 2i ✓ ◆ where (n)i =(n)(n 1)...(n i + 1) (falling factorial).
Load more

Recommended publications
  • On the Cycle Double Cover Problem
    On The Cycle Double Cover Problem Ali Ghassâb1 Dedicated to Prof. E.S. Mahmoodian Abstract In this paper, for each graph , a free edge set is defined. To study the existence of cycle double cover, the naïve cycle double cover of have been defined and studied. In the main theorem, the paper, based on the Kuratowski minor properties, presents a condition to guarantee the existence of a naïve cycle double cover for couple . As a result, the cycle double cover conjecture has been concluded. Moreover, Goddyn’s conjecture - asserting if is a cycle in bridgeless graph , there is a cycle double cover of containing - will have been proved. 1 Ph.D. student at Sharif University of Technology e-mail: [email protected] Faculty of Math, Sharif University of Technology, Tehran, Iran 1 Cycle Double Cover: History, Trends, Advantages A cycle double cover of a graph is a collection of its cycles covering each edge of the graph exactly twice. G. Szekeres in 1973 and, independently, P. Seymour in 1979 conjectured: Conjecture (cycle double cover). Every bridgeless graph has a cycle double cover. Yielded next data are just a glimpse review of the history, trend, and advantages of the research. There are three extremely helpful references: F. Jaeger’s survey article as the oldest one, and M. Chan’s survey article as the newest one. Moreover, C.Q. Zhang’s book as a complete reference illustrating the relative problems and rather new researches on the conjecture. A number of attacks, to prove the conjecture, have been happened. Some of them have built new approaches and trends to study.
    [Show full text]
  • Large Girth Graphs with Bounded Diameter-By-Girth Ratio 3
    LARGE GIRTH GRAPHS WITH BOUNDED DIAMETER-BY-GIRTH RATIO GOULNARA ARZHANTSEVA AND ARINDAM BISWAS Abstract. We provide an explicit construction of finite 4-regular graphs (Γk)k∈N with girth Γk k → ∞ as k and diam Γ 6 D for some D > 0 and all k N. For each dimension n > 2, we find a → ∞ girth Γk ∈ pair of matrices in SLn(Z) such that (i) they generate a free subgroup, (ii) their reductions mod p generate SLn(Fp) for all sufficiently large primes p, (iii) the corresponding Cayley graphs of SLn(Fp) have girth at least cn log p for some cn > 0. Relying on growth results (with no use of expansion properties of the involved graphs), we observe that the diameter of those Cayley graphs is at most O(log p). This gives infinite sequences of finite 4-regular Cayley graphs as in the title. These are the first explicit examples in all dimensions n > 2 (all prior examples were in n = 2). Moreover, they happen to be expanders. Together with Margulis’ and Lubotzky-Phillips-Sarnak’s classical con- structions, these new graphs are the only known explicit large girth Cayley graph expanders with bounded diameter-by-girth ratio. 1. Introduction The girth of a graph is the edge-length of its shortest non-trivial cycle (it is assigned to be infinity for an acyclic graph). The diameter of a graph is the greatest edge-length distance between any pair of its vertices. We regard a graph Γ as a sequence of its connected components Γ = (Γk)k∈N each of which is endowed with the edge-length distance.
    [Show full text]
  • Girth in Graphs
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMBINATORIAL THEORY. Series B 35, 129-141 (1983) Girth in Graphs CARSTEN THOMASSEN Mathematical institute, The Technicai University oJ Denmark, Building 303, Lyngby DK-2800, Denmark Communicated by the Editors Received March 31, 1983 It is shown that a graph of large girth and minimum degree at least 3 share many properties with a graph of large minimum degree. For example, it has a contraction containing a large complete graph, it contains a subgraph of large cyclic vertex- connectivity (a property which guarantees, e.g., that many prescribed independent edges are in a common cycle), it contains cycles of all even lengths module a prescribed natural number, and it contains many disjoint cycles of the same length. The analogous results for graphs of large minimum degree are due to Mader (Math. Ann. 194 (1971), 295-312; Abh. Math. Sem. Univ. Hamburg 31 (1972), 86-97), Woodall (J. Combin. Theory Ser. B 22 (1977), 274-278), Bollobis (Bull. London Math. Sot. 9 (1977), 97-98) and Hlggkvist (Equicardinal disjoint cycles in sparse graphs, to appear). Also, a graph of large girth and minimum degree at least 3 has a cycle with many chords. An analogous result for graphs of chromatic number at least 4 has been announced by Voss (J. Combin. Theory Ser. B 32 (1982), 264-285). 1, INTRODUCTION Several authors have establishedthe existence of various configurations in graphs of sufficiently large connectivity (independent of the order of the graph) or, more generally, large minimum degree (see, e.g., [2, 91).
    [Show full text]
  • Computing the Girth of a Planar Graph in Linear Time∗
    Computing the Girth of a Planar Graph in Linear Time∗ Hsien-Chih Chang† Hsueh-I Lu‡ April 22, 2013 Abstract The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an n-node unweighted undirected planar graph. The first non-trivial algorithm for the problem, given by Djidjev, runs in O(n5/4 log n) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log2 n). Weimann and Yuster further reduced the running time to O(n log n). In this paper, we solve the problem in O(n) time. 1 Introduction Let G be an edge-weighted simple graph, i.e., G does not contain multiple edges and self- loops. We say that G is unweighted if the weight of each edge of G is one. A cycle of G is simple if each node and each edge of G is traversed at most once in the cycle. The girth of G, denoted girth(G), is the minimum weight of all simple cycles of G. For instance, the girth of each graph in Figure 1 is four. As shown by, e.g., Bollobas´ [4], Cook [12], Chandran and Subramanian [10], Diestel [14], Erdos˝ [21], and Lovasz´ [39], girth is a fundamental combinatorial characteristic of graphs related to many other graph properties, including degree, diameter, connectivity, treewidth, and maximum genus. We address the problem of computing the girth of an n-node graph. Itai and Rodeh [28] gave the best known algorithm for the problem, running in time O(M(n) log n), where M(n) is the time for multiplying two n × n matrices [13].
    [Show full text]
  • The Line Mycielskian Graph of a Graph
    [VOLUME 6 I ISSUE 1 I JAN. – MARCH 2019] e ISSN 2348 –1269, Print ISSN 2349-5138 http://ijrar.com/ Cosmos Impact Factor 4.236 THE LINE MYCIELSKIAN GRAPH OF A GRAPH Keerthi G. Mirajkar1 & Veena N. Mathad2 1Department of Mathematics, Karnatak University's Karnatak Arts College, Dharwad - 580001, Karnataka, India 2Department of Mathematics, University of Mysore, Manasagangotri, Mysore-06, India Received: January 30, 2019 Accepted: March 08, 2019 ABSTRACT: In this paper, we introduce the concept of the line mycielskian graph of a graph. We obtain some properties of this graph. Further we characterize those graphs whose line mycielskian graph and mycielskian graph are isomorphic. Also, we establish characterization for line mycielskian graphs to be eulerian and hamiltonian. Mathematical Subject Classification: 05C45, 05C76. Key Words: Line graph, Mycielskian graph. I. Introduction By a graph G = (V,E) we mean a finite, undirected graph without loops or multiple lines. For graph theoretic terminology, we refer to Harary [3]. For a graph G, let V(G), E(G) and L(G) denote the point set, line set and line graph of G, respectively. The line degree of a line uv of a graph G is the sum of the degree of u and v. The open-neighborhoodN(u) of a point u in V(G) is the set of points adjacent to u. For each ui of G, a new point uˈi is introduced and the resulting set of points is denoted by V1(G). Mycielskian graph휇(G) of a graph G is defined as the graph having point set V(G) ∪V1(G) ∪v and line set E(G) ∪ {xyˈ : xy∈E(G)} ∪ {y ˈv : y ˈ ∈V1(G)}.
    [Show full text]
  • Lecture 13: July 18, 2013 Prof
    UChicago REU 2013 Apprentice Program Spring 2013 Lecture 13: July 18, 2013 Prof. Babai Scribe: David Kim Exercise 0.1 * If G is a regular graph of degree d and girth ≥ 5, then n ≥ d2 + 1. Definition 0.2 (Girth) The girth of a graph is the length of the shortest cycle in it. If there are no cycles, the girth is infinite. Example 0.3 The square grid has girth 4. The hexagonal grid (honeycomb) has girth 6. The pentagon has girth 5. Another graph of girth 5 is two pentagons sharing an edge. What is a very famous graph, discovered by the ancient Greeks, that has girth 5? Definition 0.4 (Platonic Solids) Convex, regular polyhedra. There are 5 Platonic solids: tetra- hedron (four faces), cube (\hexahedron" - six faces), octahedron (eight faces), dodecahedron (twelve faces), icosahedron (twenty faces). Which of these has girth 5? Can you model an octahedron out of a cube? Hint: a cube has 6 faces, 8 vertices, and 12 edges, while an octahedron has 8 faces, 6 vertices, 12 edges. Definition 0.5 (Dual Graph) The dual of a plane graph G is a graph that has vertices corre- sponding to each face of G, and an edge joining two neighboring faces for each edge in G. Check that the dual of an dodecahedron is an icosahedron, and vice versa. Definition 0.6 (Tree) A tree is a connected graph with no cycles. Girth(tree) = 1. For all other connected graphs, Girth(G) < 1. Definition 0.7 (Bipartite Graph) A graph is bipartite if it is colorable by 2 colors, or equiva- lently, if the vertices can be partitioned into two independent sets.
    [Show full text]
  • Dynamic Cage Survey
    Dynamic Cage Survey Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809, U.S.A. [email protected] Robert Jajcay Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809, U.S.A. [email protected] Department of Algebra Comenius University Bratislava, Slovakia [email protected] Submitted: May 22, 2008 Accepted: Sep 15, 2008 Version 1 published: Sep 29, 2008 (48 pages) Version 2 published: May 8, 2011 (54 pages) Version 3 published: July 26, 2013 (55 pages) Mathematics Subject Classifications: 05C35, 05C25 Abstract A(k; g)-cage is a k-regular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions. the electronic journal of combinatorics (2013), #DS16 1 Contents 1 Origins of the Problem 3 2 Known Cages 6 2.1 Small Examples . 6 2.1.1 (3,5)-Cage: Petersen Graph . 7 2.1.2 (3,6)-Cage: Heawood Graph . 7 2.1.3 (3,7)-Cage: McGee Graph . 7 2.1.4 (3,8)-Cage: Tutte-Coxeter Graph . 8 2.1.5 (3,9)-Cages . 8 2.1.6 (3,10)-Cages . 9 2.1.7 (3,11)-Cage: Balaban Graph . 9 2.1.8 (3,12)-Cage: Benson Graph . 9 2.1.9 (4,5)-Cage: Robertson Graph . 9 2.1.10 (5,5)-Cages .
    [Show full text]
  • Properties Ii1: Girth and Circumference
    Iernat. J. Math. & Math. Si. 685 Vol. 2 No. 4 (1979) 685-692 A GRAPH AND ITS COMPLEMENT WITH SPECIFIED PROPERTIES II1: GIRTH AND CIRCUMFERENCE JIN AKIYAMA AND FRANK HARARY Department of Mathematics The University of Michigan Ann Arbor, Michigan 48109 U.S.A. (Received April 5, 1979) ABSTRACT. In this series, we investigate the conditions under which both a graph G and its complement G possess certain specified properties. We now characterize all the graphs G such that both G and G have the same girth. We also determine all G such that both G and G have circumference 3 or4. KEF WORDS AND PHRASES. Graph, Complement, Girth, bmfence. 1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. 05C99. iVisiting Scholar, 1978-79, from Nippon Ika University, Kawasaki, Japan. 2Vice-President, Calcutta Mathematical Society, 1978 and 1979. 686 J. AKIYAMA AND F. HARARY I. NOTATIONS AND BACKGROUND. In the first paper [2] in this series, we found all graphs G such that both G and its complement G have (a) connectivity 1, (b) line-connectivity 1, (c) mo cycles, (d) only even cycles, and other properties. Continuing this study, we determined in [3] the graphs G for which G and 6 are (a) block-graphs, (b) middle graphs, () bivariegated, and (d) n'th subdivision graphs. Now we concentrate on the two invariants concerning cycle lengths- girth and circum- ference. We will see that whenever G and G have the same girth g then g 3 or 5 only. For the circumference c we study only the cases where both G and G have c 3 or 4 Following the notation and of the G + G of two terminology [4], join 1 2 graphs is the union of G and G with the sets 1 2 complete bigraph having point V and V We will require a related ternary operation denoted G + G + G 1 2 1 2 3 on three disjoint graphs, defined as the union of the two joins G + G and 1 2 G + G Thus, this resembles the composition of the path not with just 2 3 P3 one other graph but with three graphs, one for each point; Figure 1 illustrates the "random" graph K e K + K + K Of course the quaternary operation 4 1 2 1 G + G + G + G is defined similarly.
    [Show full text]
  • Computing the Girth of a Planar Graph in O(Nlogn) Time
    Computing the girth of a planar graph in O(n log n) time Oren Weimann and Raphael Yuster ICALP’2009 & SIAM Journal on Discrete Mathematics 24 (2010) 609–616. Speaker: Joseph, Chuang-Chieh Lin Supervisor: Professor Maw-Shang Chang Computation Theory Laboratory Department of Computer Science and Information Engineering National Chung Cheng University, Taiwan December 7, 2010 1/38 Outline 1 Introduction 2 Planar graphs and k-outerplanar graphs The face size & the girth General ideas of the O(n log n) algorithm 3 The divide-and-conquer algorithm for k-outerplanar graphs 2/38 Outline 1 Introduction 2 Planar graphs and k-outerplanar graphs The face size & the girth General ideas of the O(n log n) algorithm 3 The divide-and-conquer algorithm for k-outerplanar graphs 3/38 Girth Definition (The girth of a graph G) The length of the shortest cycle of G. The girth has tight connections to many graph properties. chromatic number; minimum or average vertex-degree; diameter; connectivity; genus; ... 4/38 Girth Definition (The girth of a graph G) The length of the shortest cycle of G. The girth has tight connections to many graph properties. chromatic number; minimum or average vertex-degree; diameter; connectivity; genus; ... 4/38 The road of computing the girth of a graph For general graphs G = (V , E), n = V and m = E : | | | | O(nm) [Itai & Rodeh, SIAM J. Comput. 1978]. O(n2) with an additive error of one. For computing the shortest even-length cycle: O(n2α(n)) [Monien, Computing 1983]. O(n2) [Yuster & Zwick, SIAM J.
    [Show full text]
  • COLORING GRAPHS with FIXED GENUS and GIRTH 1. Introduction Grötzsch
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number 11, November 1997, Pages 4555{4564 S 0002-9947(97)01926-0 COLORING GRAPHS WITH FIXED GENUS AND GIRTH JOHN GIMBEL AND CARSTEN THOMASSEN Abstract. It is well known that the maximum chromatic number of a graph 1=2 on the orientable surface Sg is θ(g ). We prove that there are positive con- stants c1;c2 such that every triangle-free graph on Sg has chromatic number 1=3 less than c2(g= log(g)) and that some triangle-free graph on Sg has chro- g1=3 matic number at least c1 log(g) . We obtain similar results for graphs with restricted clique number or girth on Sg or Nk. As an application, we prove 3=7 that an Sg-polytope has chromatic number at most O(g ). For specific sur- faces we prove that every graph on the double torus and of girth at least six is 3-colorable and we characterize completely those triangle-free projective graphs that are not 3-colorable. 1. Introduction Gr¨otzsch [14] proved that every planar graph with no triangles can be 3-colored. A short proof is given in [23]. Kronk and White [18] proved that every toroidal graph with no triangles can be 4-colored and that every toroidal graph with no cycles of length less than six can be 3-colored. Kronk [17] studied the chromatic number of triangle-free graphs on certain surfaces. Thomassen [23] showed that every graph on the torus with girth at least five is 3-colorable (as conjectured in [18]) and in the same work showed that a graph which embeds on the projective plane with no contractible 3-cycle nor 4-cycle is 3-colorable.
    [Show full text]
  • Bounds on Graphs with High Girth and High Chromatic Number
    Graph Theory Erdos˝ Results Bounds on graphs with high girth and high chromatic number Enrique Treviño joint work with Daniel Bath and Zequn Li INTEGERS 2013: The Erdos˝ Centennial Conference October 26, 2013 Enrique Treviño Bounds on graphs with high girth and high chromatic number Graph Theory Erdos˝ Results Some Definitions Chromatic Number: The chromatic number of a graph G is the smallest number of colors needed to color the vertices of G so that no two adjacent vertices share the same color. Cycle: A cycle graph of length n is an n sided polygon (i.e., a graph with n vertices and n edges where each vertex has degree 2). Example: Enrique Treviño Bounds on graphs with high girth and high chromatic number Graph Theory Erdos˝ Results More Definitions Independent Set: An independent set in a graph is a set of vertices no two of which are adjacent. Girth Number : The girth number of a graph is the length of a shortest cycle contained in the graph. If the graph does not contain any cycles (i.e. it’s an acyclic graph), its girth is defined to be infinity. In the following example the girth number is 8: Enrique Treviño Bounds on graphs with high girth and high chromatic number Graph Theory Erdos˝ Results Examples Figure: The maximal independent set consists of vertices f8; 1; 3; 5g, the chromatic number is at least 5 because the vertices f2; 8; 7; 4; 6g form a complete graph. The girth number is 3 because 3 is the smallest cycle. Enrique Treviño Bounds on graphs with high girth and high chromatic number Graph Theory Erdos˝ Results Erdos,˝ 1959 In the paper “Graph Theory and Probability”, Erdos˝ proved the following: Let h(k; l) be the least integer such that every graph with h(k; l) vertices contains either a cycle of k or fewer edges or the graph contains a set of l independent vertices.
    [Show full text]
  • On Approximating the D-Girth of a Graph
    On approximating the d-girth of a graph David Peleg1, Ignasi Sau2, and Mordechai Shalom3 1 Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel. [email protected] 2 AlGCo project-team, CNRS, Laboratoire d'Informatique de Robotique et de Micro¶electroniquede Montpellier (LIRMM), Montpellier, France. [email protected] 3 TelHai Academic College, Upper Galilee, 12210, Israel. [email protected] Abstract. For a ¯nite, simple, undirected graph G and an integer d ¸ 1, a mindeg-d subgraph is a subgraph of G of minimum degree at least d. The d- girth of G, denoted gd(G), is the minimum size of a mindeg-d subgraph of G. It is a natural generalization of the usual girth, which coincides with the 2-girth. The notion of d-girth was proposed by Erd}os et al. [13, 14] and Bollob¶asand Brightwell [7] over 20 years ago, and studied from a purely combinatorial point of view. Since then, no new insights have appeared in the literature. Recently, ¯rst algorithmic studies of the problem have been carried out [2,4]. The current article further explores the complexity of ¯nding a small mindeg-d subgraph of a given graph (that is, approximating its d-girth), by providing new hardness results and the ¯rst approximation algorithms in general graphs, as well as analyzing the case where G is planar. Keywords: generalized girth, minimum degree, approximation algorithm, hard- ness of approximation, randomized algorithm, planar graph. 1 Introduction Degree-constrained subgraph problems have attracted considerable attention in the last decades, resulting in a large body of literature (see e.g.
    [Show full text]