DOCSLIB.ORG

On the Linear K-Arboricity of Cubic Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Linear K-Arboricity of Cubic Graphs On the Linear k-arboricity of Cubic Graphs Bill Jackson Department of Mathematical Studies Goldsmiths' College London SE14 6NW, England Nicholas C. Wormald Department of Mathematics University of Melbourne Parkville, VIC 3052 Australia Abstract Bermond et al. [2] conjectured that the edge set of a cubic graph G can be partitioned into two linear k-forests, that is to say two forests whose connected components are paths of length at most k, for all k 5. We shall prove a weaker result that the statement is valid for ¸ all k 18. ¸ All graphs considered will be ¯nite. We shall refer to graphs which may contain loops or multiple edges as multigraphs and reserve the term graph for those which do not. A linear forest is a forest each of whose components are paths. The linear arboricity of a graph G, de¯ned by Harary [7], is the minimum number of linear forests required to partition E(G) and is denoted by la(G). It was shown by Akiyama, Exoo and Harary [1] that la(G) = 2 when G is cubic. A k-linear forest is a forest consisting of paths of length at most k. The k-linear arboricity of G, introduced by Bermond et al. [2], is the minimum number of k-linear forests required to partition E(G), and is denoted by lak(G). It is conjectured in [2] that if G is cubic then la5(G) 2. A partial result is obtained by Delamarre et al. [3] who show that la (G)· 2 k · 1 1 when k 2 V (G) 4. The purpose of this note is to improve their result, when V¸(G)j 36j, ¸to: j j ¸ Theorem. Let G be a cubic graph and k 18 be an integer. Then lak(G) = 2. ¸ In order to prove this theorem we shall need the following lemma on matchings in multigraphs. Lemma. Let H be a multigraph with m edges and maximum degree at most four. Let x be the number of loops in components with only one vertex, and q the number of loops incident with vertices of degree 3. Then H has a matching of size at least (2m 2x q)=12. ¡ ¡ Proof. We may assume that H is connected and hence x = 0. By the Gallai-Edmonds Theorem [5, 6, 4] there exists a partition R; S; T of V (H) such that: f g H has no edges from R to T ; ² each component of H[R] has a perfect matching; ² each component of H[T ] is matching critical (i.e. the deletion of any ² vertex leaves a graph with a perfect matching); the maximum size of a matching in H, f, is given by ² f = r=2 + s + (t w)=2; ¡ where r = R , s = S , t = T and w is the number of components of H[T ]. j j j j j j Let w0 and w1 be the number of components of H[T ] which have, re- spectively, zero edges and one vertex, and one edge and one vertex. Since each vertex of H has degree at most four E(H[R]) 2r, and the num- ber of edges of H incident with vertices in jS is at mostj · 4s. Also, since H is connected, each component of H[T ] is adjacent to a vertex of S and so E(H[T ]) 2t w w . Thus j j · ¡ ¡ 0 m 2r + 4s + 2t w w : · ¡ ¡ 0 2 Again using the fact that H is connected we deduce that the number of edges from T to S is at least w + (w1 q). On the other hand, it is at most 4s, so 4s w + w q. Thus w 4s¡ w + q and ¸ 1 ¡ 1 · ¡ t 3w 2w 2w 5w 2w 8s 2q: ¸ ¡ 0 ¡ 1 ¸ ¡ 0 ¡ ¡ Now 12f 6r + 12s + 6t 6w ¸ ¡ 2m + 4s + 2t 4w + 2w ¸ ¡ 0 2m + 4s + 3t=2 + (5w 2w 8s 2q)=2 4w + 2w ¸ ¡ 0 ¡ ¡ ¡ 0 = 2m q + 3t=2 3w=2 + w ¡ ¡ 0 2m q ¸ ¡ since t w. ¸ Proof of Theorem. De¯ne = (G) to be the set of (F ; F ) such that F F 1 2 F1 and F2 are two linear forests whose edges partition the edges of G. Then is nonempty by Akiyama et al. [1]. For (F ; F ) , let l(F ; F ) denote F 1 2 2 F 1 2 the length of the longest path in F1 or F2. Let t be the minimum number such that there exists (F1; F2) with l(F1; F2) = t. Suppose that t 4. We will2 Fde¯ne sets S ; S ; : : : of paths of G, and ¸ 0 1 disjoint sets T0; T1; : : : of edges of G. We shall show that if t 19 then T = for all i 0 , thus contradicting the ¯niteness of G. ¸ i 6 ; ¸ We will choose Si and Ti such that Ti is a subset of the internal edges of the paths in S , and such that for some (F ; F ) with l(F ; F ) = t, the i 1 2 2 F 1 2 paths in Si are in F1 if i is even, and in F2 if i is odd. Given path sets S0; : : : ; Si, de¯ne Ri = Ri(S0; : : : ; Si) to be the set of all edges of G which are either contained in or incident with any of the paths in S0; : : : ; Si. Given (F ; F ) and path sets S ; : : : ; S , we say that (F 0; F 0) is 1 2 2 F 0 i 1 2 2 F an i-improvement of (F1; F2) if there exists an edge e of a path in Si such that (i) all edges in R e are in F 0 i® they are in F , i n 1 1 (ii) e is in F10 i® it is not in F1, and (iii) each path in Fj0 of length at least t is also a path in Fj, j = 1 and 2. 3 We will de¯ne the sets S and T inductively. First choose (F ; F ) i i 1 2 2 F with l(F1; F2) = t and such that, subject to this condition, the total number, m(F1; F2), of paths of length t in F1 and F2 is as small as possible. By symmetry we may assume that F has a path P of length t. Let S = P . 1 0 f g Let T0 be a maximum set of pairwise non-adjacent internal edges in P such that for each path in F2, only one end of the path is incident with an edge in T0. Note that T0 is nonempty because t 4. Inductively for i > 0 carry out the following¸ two steps. 1 Repeat (a) and (b) consecutively until (b) requires no action because its condition fails: (a) De¯ne Si to be the set of paths in F1 (if i is even) or F2 (if i is odd) which terminate at vertices incident with edges in Ti 1. ¡ (b) If there is an i-improvement (F10; F20) of (F1; F2) then reset (F1; F2) to be equal to (F10; F20). 2 De¯ne Ti to be a maximum set of pairwise non-adjacent internal edges in the paths in Si such that neither path in Fj (where j is 1 or 2 if i is odd or even respectively) terminating at such an edge is in Sl for any 0 l < i, and such that no path in F has both endvertices incident · j with edges in Ti. Step 1 must eventually terminate because the rede¯nition of (F1; F2) in 1(b) leads to a strictly smaller total length of paths in the set Si when 1(a) is next performed. For i 0 let si = Si and ti = Ti . It follows from Steps 1(a) and 2 that ¸ j j j j si = 2ti 1 (1) ¡ for all i 1. ¸ Denote the value of (F1; F2) at the end of the ith iteration of the two steps above by (Fi;1; Fi;2). Then for any j > i the set Si retains its property of being a set of maximal paths in Fj;1 or Fj;2, because the rede¯nition of (F1; F2) does not disturb any edges in Rj 1. Thus, (Fj;1; Fj;2) cannot have ¡ any i-improvement for i < j. Provided i 2, consider any edge e in Ti 1 and the two paths P1 and ¸ ¡ P2 in Si which terminate at the endvertices of e. For the sake of argument we can assume e F . Then the total length of P and P must be at least 2 1 1 2 4 t 1; otherwise we could put Fi0 1;1 = Fi 1;1 e and Fi0 1;2 = Fi 1;2 + e to ¡ ¡ ¡ ¡ ¡ ¡ obtain an (i 1)-improvement of (Fi 1;1; Fi 1;2). A similar argument works ¡ ¡ if i = 1, although¡ in that case the contradiction reached is that the number of paths of length t is reduced, as P gets destroyed. We can now deduce that for i 1 the total length of paths in Si is at least (t 1)ti 1. Thus by (1) ¸ ¡ ¡ there are at least si(t 5)=2 internal edges in the paths in Si. We now show that¡T = for all i 0. We have already seen that this i 6 ; ¸ holds for i = 0. Hence suppose that i > 0. Let Xi be the subgraph of G induced by the internal edges of the paths in Si. Let wi be the number of paths in some Sj, 0 j < i, with an end-vertex in Xi.
Load more

Recommended publications
  • Visualizing Graphs: Optimization and Trade-Offs
    Visualizing Graphs: Optimization and Trade-offs by Debajyoti Mondal A Thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfilment of the requirements of the degree of DOCTOR OF PHILOSOPHY Department of Computer Science University of Manitoba Winnipeg Copyright c 2016 by Debajyoti Mondal Thesis advisor Author Dr. Stephane Durocher Debajyoti Mondal Abstract Effective visualization of graphs is a powerful tool to help understand the rela- tionships among the graph’s underlying objects and to interact with them. Sev- eral styles for drawing graphs have emerged over the last three decades. Polyline drawing is a widely used style for drawing graphs, where each node is mapped to a distinct point in the plane and each edge is mapped to a polygonal chain between their corresponding nodes. Some common optimization criteria for such a drawing are defined in terms of area requirement, number of bends per edge, angular resolution, number of distinct line segments, edge crossings, and number of planar layers. In this thesis we develop algorithms for drawing graphs that optimize different aesthetic qualities of the drawing. Our algorithms seek to simultaneously opti- mize multiple drawing aesthetics, reveal potential trade-offs among them, and improve many previous graph drawing algorithms. We start by exploring probable trade-offs in the context of planar graphs. We prove that every n-vertex planar triangulation G with maximum degree D can be drawn with at most 2n + t 3 segments and O(8t D2t) area, where t is the − · number of leaves in a Schnyder tree of G.
    [Show full text]
  • Arxiv:1712.05006V1 [Math.CO] 13 Dec 2017 Oebr2,2017
    The List Linear Arboricity of Graphs Ringi Kim1 Department of Mathematical Sciences KAIST Daejeon South Korea 34141 and Luke Postle2 Department of Combinatorics and Optimization University of Waterloo Waterloo, ON Canada N2L 3G1 ABSTRACT A linear forest is a forest in which every connected component is a path. The linear arboricity of a graph G is the minimum number of linear forests of G covering all edges. In 1980, Akiyama, Exoo and Harary proposed a conjecture, known as the Linear Arboricity Conjecture (LAC), stating ∆+1 that every ∆-regular graph G has linear arboricity 2 . In 1988, Alon proved that the LAC holds asymptotically. In 1999, the list version of the LAC wa s raised by An and Wu, which is called the List Linear Arboricity Conjecture. In this article, we prove that the List Linear Arboricity Conjecture holds asymptotically. arXiv:1712.05006v1 [math.CO] 13 Dec 2017 November 23, 2017. [email protected]. [email protected]. This research was partially supported by NSERC under Discovery Grant No. 2014- 06162, the Ontario Early Researcher Awards program and the Canada Research Chairs program. 1 1 Introduction In this paper, we consider only undirected simple graphs. A linear forest is a forest in which every connected component is a path. The linear arboricity of a graph G, denoted by la(G), first introduced by Harary [13], is the minimum number of linear forests of G needed to cover all edges of G. Akiyama, Exoo and Harary [1] proposed a conjecture, known as the Linear Arboricity ∆+1 Conjecture, stating that for every ∆-regular graph G, la(G) = 2 .
    [Show full text]
  • 2017Sacls104
    2017SACLS104 THÈSE DE DOCTORAT DE SHANDONG UNIVERSITY ET DE L’UNIVERSITÉ PARIS-SACLAY PRÉPARÉE À L’UNIVERSITÉ PARIS-SUD ÉCOLE DOCTORALE N°580 Sciences et technologies de l’information et de la communication GRADUATE SCHOOL of Mathematics Spécialité Informatique Par Mme Renyu Xu Quelques problèmes de coloration du graphe Thèse présentée et soutenue à Jinan, le 27 Mai 2017 : Composition du Jury : M. Hongjian Lai, Professeur, West Virginia University, Présidente du Jury Mme Lianying Miao, Professeure, China University of Mining and Technology, Rapporteur Mme Yinghong Ma, Professeure, Shandong Normal University, Rapporteur M. Evripidis Bampis, Professeur, Université Paris VI, Rapporteur Mme Michèle Sebag, Directrice de Recherche CNRS, Examinatrice M. Yannis Manoussakis, Professeur, Directeur de thèse M. Jianliang Wu, Professeur, CoDirecteur de thèse Contents Abstract iii Re´sume´(Abstract in French) vii Symbol Description xiii Chapter 1 Introduction 1 1.1 Basic Definitions and Notations . 3 1.2 SomeSpecialgraphs ....................... 4 1.3 Somecoloringproblemsofgraphs . 7 1.4 Mainresults............................ 21 Chapter 2 Total Coloring 25 2.1 Basic definitions and properties . 25 2.2 Planar graph G with maximum degree ∆ ≥ 8 ......... 25 2.2.1 If every 6-cycle of G contains at most one chord or chordal 6-cycles are not adjacent in G .......... 26 2.3.1 7-cycles containing at most two chords . 38 2.5 Planar graph with ∆ ≥ 7..................... 53 Chapter 3 List Coloring 65 3.1 Listvertexcoloring ........................ 65 3.2 List edge coloring and List total coloring . 68 3.2.1 Planar graph G with maximum degree ∆ ≥ 8...... 69 3.2.2 Planar graph G with maximum degree ∆ ≥ 6.....
    [Show full text]
  • The Geometric Thickness of Low Degree Graphs
    The Geometric Thickness of Low Degree Graphs CHRISTIAN A. DUNCAN DAVID EPPSTEIN∗ Department of Computer Science Department of Computer Science University of Miami University of California, Irvine [email protected] [email protected] STEPHEN G. KOBOUROV† Department of Computer Science University of Arizona [email protected] Abstract We prove that the geometric thickness of graphs whose maximumdegreeis no more than four is two. All of our algorithms run in O(n) time, where n is the number of vertices in the graph. In our proofs, we present an embedding algorithm for graphs with maximum degree three that uses an n n grid and a more complex algorithm for embedding a graph with maximum degree four. We also show× a variation using orthogonal edges for maximum degree four graphs that also uses an n n grid. The results have implications in graph theory, graph drawing, and VLSI design. × 1 Introduction The thickness of a graph G = (V,E) is the smallest number of planar subgraphs needed to decompose G. That is, we wish to find the smallest number i such that we can partition the edges E into i different planar subgraphs; see [17] for a survey. Using the terminology from VLSI, these subgraphs are referred to as the layers of the original graph. In several applications, including VLSI layouts [1], visualization of software development [10], and graph drawing [15] it is necessary that the vertex locations be consistent across the arXiv:cs/0312056v1 [cs.CG] 24 Dec 2003 layers. It is well known that a planar graph can be drawn in the plane without crossings, using arbitrary placement of the vertices and Jordan curves representing the edges.
    [Show full text]
  • Drawing Graphs on Few Lines and Few Planes∗
    Journal of Computational Geometry jocg.org DRAWING GRAPHS ON FEW LINES AND FEW PLANES∗ Steven Chaplick,yz Krzysztof Fleszar,§ Fabian Lipp,y{ Alexander Ravsky,k Oleg Verbitsky,∗∗ and Alexander Wolffyyy Abstract. We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight- line edges and crossing-free drawings. This problem has many relations to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track draw- ings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows. – We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. – We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments ap- pearing in a drawing). – We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on asymptotically fewer lines than in the plane. 1 Introduction It is well known that any graph admits a crossing-free straight-line drawing in 3-space.
    [Show full text]
  • Orientation-Based Edge-Colorings and Linear Arboricity of Multigraphs
    Orientation-based edge-colorings and linear arboricity of multigraphs Ronen Wdowinski∗ August 27, 2021 Abstract The Goldberg-Seymour Conjecture for f-colorings states that the f-chromatic index of a loopless multigraph is essentially determined by either a maximum degree or a maximum density parameter. We introduce an oriented version of f-colorings, where now each color class of the edge-coloring is required to be orientable in such a way that every vertex v has indegree and outdegree at most some specified values g(v) and h(v). We prove that the associated (g, h)-oriented chromatic index satisfies a Goldberg- Seymour formula. We then present simple applications of this result to variations of f-colorings. In particular, we show that the Linear Arboricity Conjecture holds for k- degenerate loopless multigraphs when the maximum degree is at least 4k−2, improving a bound recently announced by Chen, Hao, and Yu for simple graphs. Finally, we demonstrate that the (g, h)-oriented chromatic index is always equal to its list coloring analogue. 1 Introduction For basic terminology on multigraphs, simple graphs, and oriented graphs, see Diestel [11]. arXiv:2108.11816v1 [math.CO] 26 Aug 2021 1.1 f-colorings Given a loopless multigraph G and a function f : V (G) → N\{0}, an f-coloring of G is an assignment of a color to each edge of G such that each color class is a subgraph of G in which every vertex v ∈ V (G) has degree d(v) at most f(v). We will refer to such subgraphs as degree-f subgraphs.
    [Show full text]
  • Arxiv:1607.01196V2 [Cs.CG] 1 Sep 2016 Pose That We Are Allowed to Draw Edges Only on a Limited Number of Planes
    Drawing Graphs on Few Lines and Few Planes? Steven Chaplick1, Krzysztof Fleszar1, Fabian Lipp1, Alexander Ravsky2, Oleg Verbitsky3, and Alexander Wolff1 1 Lehrstuhl f¨urInformatik I, Universit¨atW¨urzburg,Germany, www1.informatik.uni-wuerzburg.de/en/staff 2 Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Science of Ukraine, Lviv, Ukraine 3 Institut f¨urInformatik, Humboldt-Universit¨atzu Berlin, Germany Abstract. We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows. { We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. { We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, maximum de- gree, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments appearing in a drawing). { We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on a lot fewer lines than in the plane.
    [Show full text]
  • A Planar Linear Arboricity Conjecture
    A Planar Linear Arboricity Conjecture∗ Marek Cygan1,Lukasz Kowalik1, Borut Luˇzar2† 1 Institute of Informatics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland {cygan,kowalik}@mimuw.edu.pl 2 Institute of Mathematics, Physics, and Mechanics, Jadranska 19, 1111 Ljubljana, Slovenia [email protected] Abstract The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1984, Akiyama et al. [1] stated the Linear Arboricity Conjecture (LAC), that the linear arboricity of any simple graph of maximum degree ∆ ∆ ∆+1 is either 2 or 2 . In [14, 17] it was proven that LAC holds for all planar graphs. ∆ LAC implies that for ∆ odd, la(G)= 2 . We conjecture that for planar graphs this equality is true also for any even ∆ ≥ 6. In this paper we show that it is true for any even ∆ ≥ 10, leaving open only the cases ∆ = 6, 8. We present also an O(n log n) algorithm for partitioning a planar graph into max{la(G), 5} linear forests, which is optimal when ∆ ≥ 9. 1 Introduction In this paper we consider only undirected and simple graphs. A linear forest is a forest in which every connected component is a path. The linear arboricity la(G) of a graph G is the minimum number of linear forests in G, whose union is the set of all edges of G. This one of the most natural graph covering notions was introduced by Harary [9] in 1970. Note that for ∆ any graph of maximum degree ∆ one needs at least ⌈ 2 ⌉ linear forests to cover all the edges.
    [Show full text]
  • A Status on the Linear Arboricity J
    A STATUS ON THE LINEAR ARBORICITY J. Akiyama Department of Mathematics, Nippon Ika University Kawasaki, Japan 211 Abstract. In a linear forest, each component is a path. The linear arboricity ~(G) of a graph G is defined in Harary [8] as the minimum number of linear forests whose union is G. This invariant first arose in a study [i0] of information retrieval in file systems. A quite similar covering invariant which is well known to the linear arboricity is the arboricity of a graph, which is defined as the minimum number of forests whose union is G. Nash-Williams [il] determined the arboricity of any graph, however only few results on the linear arbmricity are known. We shall present these discoveries and an open problem on this new invariant. i. £ntroduction In a linear forestr each component is a path° The linear arboricity ~(G) of a graph G is defined as the minimum number of linear forests whose union is G. All other defini£ions and texTainology employed in this paper can be found in Behzad, Chartrand and Lesniak-Foster [6] or Harary [9]. We now present a few fundamental results for specified families of graphs. Theorem i. If T is a tree with maximum degree ATr then (i) ~(T) = {AT/2}. Proof. The lower bound ~(T) ~ {AT/2} is obvious. Since tree T has maximum degree AT, its edge chromatic number X' (T) is equal to AT. Each subgraph induced by subsets of edges with two colors is a linear forest. Thus we obtain the upper bound: ~T) ~ {X' (T)/2 = {AT/2}.
    [Show full text]
  • On the Linear K-Arboricity of Regular Graphs
    More on the Linear k-arb oricity of Regular Graphs R. E. L. Aldred Department of Mathematics and Statistics University of Otago P.O. Box 56, Dunedin New Zealand Nicholas C. Wormald Department of Mathematics University of Melb ourne Parkville, VIC 3052 Australia Abstract Bermond et al. [2] conjectured that the edge set of a cubic graph G can b e partitioned into two linear k -forests, that is to saytwo forests whose connected comp onents are paths of length at most k , for all k 5. That the statement is valid for all k 18 was shown in [5 ] by Jackson and Wormald. Here we improve this b ound to 0 7 if G=3; k 9 otherwise. The result is also extended to d-regular graphs for d > 3, at the exp ense of increasing the numb er of forests to d 1. All graphs considered will b e nite. We shall refer to graphs which may contain lo ops or multiple edges as multigraphs and reserve the term graph for those which do not. A linear forest is a forest each of whose comp onents 1 are paths. The linear arboricity of a graph G, de ned by Harary [4], is the minimum numb er of linear forests required to partition E G and is denoted by laG. It was shown by Akiyama, Exo o and Harary [1] that laG = 2 when G is cubic. A linear k -forest is a forest consisting of paths of length at most k . The linear k -arboricity of G, intro duced by Bermond et al.
    [Show full text]
  • Notes By: Andrew Lin
    The Probabilistic Method in Combinatorics Lecturer: Professor Yufei Zhao Notes by: Andrew Lin Spring 2019 This is an edited transcript of the lectures of MIT’s Spring 2019 class 18.218: The Probabilistic Method in Combinatorics, taught by Professor Yufei Zhao. Each section focuses on a different technique, along with examples of applications. Additional course material, including problem sets, can be found on the course website. The main reference for the material is the excellent textbook N. Alon and J. H. Spencer, The probabilistic method, Wiley, 4ed. Most of the course will follow the textbook, though some parts will differ. Please contact Yufei Zhao ([email protected]) and Andrew Lin ([email protected]) for any questions or comments regarding these notes. Special thanks to Abhijit Mudigonda, Mihir Singhal, Andrew Gu, and others for their help in proofreading. Contents 1 Introduction to the probabilistic method 4 1.1 The Ramsey numbers ........................................... 4 1.2 Alterations ................................................. 5 1.3 Lovász Local Lemma ........................................... 6 1.4 Set systems ................................................ 7 1.5 Hypergraph colorings ........................................... 9 2 Linearity of expectation 11 2.1 Setup and basic examples ......................................... 11 2.2 Sum-free sets ............................................... 11 2.3 Cliques ................................................... 12 2.4 Independent sets ............................................
    [Show full text]
  • The Linear Arboricity of the Schrijver Graph SG(2K+2,K)
    BULLETIN of the Bull. Malays. Math. Sci. Soc. (2) 35(2) (2012), 239–246 MALAYSIAN MATHEMATICAL SCIENCES SOCIETY http://math.usm.my/bulletin The Linear Arboricity of the Schrijver Graph SG(2k + 2;k) 1BING XUE AND 2LIANCUI ZUO 1School of Mathematics, Shandong University, Jinan, 250100, P. R. China 2College of Mathematical Science, Tianjin Normal University, Tianjin, 300387, P. R. China [email protected], [email protected] Abstract. The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edge set E(G) of G. The vertex linear arboricity vla(G) of a graph G is the minimum number of subsets into which the vertex set V(G) can be partitioned so that every subset induces a linear forest. The Schrijver graph SG(n;k) is the graph whose vertex set consists of all 2-stable k-subsets of the set [n] = f0;1;:::;n − 1g and two vertices A and B are adjacent if and only if A \ B = f. In this paper, it is proved that la(SG(2k + 2;k)) = d(k + 2)=2e for k ≥ 3 and vla(SG(2k + 2;k)) = va(SG(2k + 2;k)) = 2 for k ≥ 2. 2010 Mathematics Subject Classification: 05C15 Keywords and phrases: Linear forest, linear arboricity, vertex linear arboricity, Schrijver graph. 1. Introduction Throughout this paper, all graphs considered are finite, undirected and simple. For a real number x, dxe is the least integer not less than x, and bxc is the most integer not more than x. For a graph G, we use V(G);E(G);D(G) to denote the vertex set, the edge set and the maximum degree, respectively.
    [Show full text]