Minors in Graphs with High Chromatic Number
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On Treewidth and Graph Minors
On Treewidth and Graph Minors Daniel John Harvey Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy February 2014 Department of Mathematics and Statistics The University of Melbourne Produced on archival quality paper ii Abstract Both treewidth and the Hadwiger number are key graph parameters in structural and al- gorithmic graph theory, especially in the theory of graph minors. For example, treewidth demarcates the two major cases of the Robertson and Seymour proof of Wagner's Con- jecture. Also, the Hadwiger number is the key measure of the structural complexity of a graph. In this thesis, we shall investigate these parameters on some interesting classes of graphs. The treewidth of a graph defines, in some sense, how \tree-like" the graph is. Treewidth is a key parameter in the algorithmic field of fixed-parameter tractability. In particular, on classes of bounded treewidth, certain NP-Hard problems can be solved in polynomial time. In structural graph theory, treewidth is of key interest due to its part in the stronger form of Robertson and Seymour's Graph Minor Structure Theorem. A key fact is that the treewidth of a graph is tied to the size of its largest grid minor. In fact, treewidth is tied to a large number of other graph structural parameters, which this thesis thoroughly investigates. In doing so, some of the tying functions between these results are improved. This thesis also determines exactly the treewidth of the line graph of a complete graph. This is a critical example in a recent paper of Marx, and improves on a recent result by Grohe and Marx. -
Average Degree Conditions Forcing a Minor∗
Average degree conditions forcing a minor∗ Daniel J. Harvey David R. Wood School of Mathematical Sciences Monash University Melbourne, Australia [email protected], [email protected] Submitted: Jun 11, 2015; Accepted: Feb 22, 2016; Published: Mar 4, 2016 Mathematics Subject Classifications: 05C83 Abstract Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger's Conjecture, much research has focused on the av- erage degree required to force a complete graph as a minor. Subsequently, various authors have considered the average degree required to force an arbitrary graph H as a minor. Here, we strengthen (under certain conditions) a recent result by Reed and Wood, giving better bounds on the average degree required to force an H-minor when H is a sparse graph with many high degree vertices. This solves an open problem of Reed and Wood, and also generalises (to within a constant factor) known results when H is an unbalanced complete bipartite graph. 1 Introduction Mader [13, 14] first proved that high average degree forces a given graph as a minor1. In particular, Mader [13, 14] proved that the following function is well-defined, where d(G) denotes the average degree of a graph G: f(H) := inffD 2 R : every graph G with d(G) > D contains an H-minorg: Often motivated by Hadwiger's Conjecture (see [18]), much research has focused on f(Kt) where Kt is the complete graph on t vertices. For 3 6 t 6 9, exact bounds on the number of edges due to Mader [14], Jørgensen [6], and Song and Thomas [19] implyp that f(Kt) = 2t−4. -
The Hadwiger Number, Chordal Graphs and Ab-Perfection Arxiv
The Hadwiger number, chordal graphs and ab-perfection∗ Christian Rubio-Montiel [email protected] Instituto de Matem´aticas, Universidad Nacional Aut´onomade M´exico, 04510, Mexico City, Mexico Department of Algebra, Comenius University, 84248, Bratislava, Slovakia October 2, 2018 Abstract A graph is chordal if every induced cycle has three vertices. The Hadwiger number is the order of the largest complete minor of a graph. We characterize the chordal graphs in terms of the Hadwiger number and we also characterize the families of graphs such that for each induced subgraph H, (1) the Hadwiger number of H is equal to the maximum clique order of H, (2) the Hadwiger number of H is equal to the achromatic number of H, (3) the b-chromatic number is equal to the pseudoachromatic number, (4) the pseudo-b-chromatic number is equal to the pseudoachromatic number, (5) the arXiv:1701.08417v1 [math.CO] 29 Jan 2017 Hadwiger number of H is equal to the Grundy number of H, and (6) the b-chromatic number is equal to the pseudo-Grundy number. Keywords: Complete colorings, perfect graphs, forbidden graphs characterization. 2010 Mathematics Subject Classification: 05C17; 05C15; 05C83. ∗Research partially supported by CONACyT-Mexico, Grants 178395, 166306; PAPIIT-Mexico, Grant IN104915; a Postdoctoral fellowship of CONACyT-Mexico; and the National scholarship programme of the Slovak republic. 1 1 Introduction Let G be a finite graph. A k-coloring of G is a surjective function & that assigns a number from the set [k] := 1; : : : ; k to each vertex of G.A k-coloring & of G is called proper if any two adjacent verticesf haveg different colors, and & is called complete if for each pair of different colors i; j [k] there exists an edge xy E(G) such that x &−1(i) and y &−1(j). -
Some Remarks on Even-Hole-Free Graphs
Some remarks on even-hole-free graphs Zi-Xia Song∗ Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Abstract A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisimplicial vertex. Both Hadwiger’s conjecture and the Erd˝os-Lov´asz Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that for all k ≥ 7, every even-hole-free graph with no Kk minor is (2k − 5)-colorable; every even-hole-free graph G with ω(G) < χ(G) = s + t − 1 satisfies the Erd˝os-Lov´asz Tihany conjecture provided that t ≥ s>χ(G)/3. Furthermore, we prove that every 9-chromatic graph G with ω(G) ≤ 8 has a K4 ∪ K6 minor. Our proofs rely heavily on the structural result of Chudnovsky and Seymour on even-hole-free graphs. 1 Introduction All graphs in this paper are finite and simple. For a graph G, we use V (G) to denote the vertex set, E(G) the edge set, |G| the number of vertices, e(G) the number of edges, δ(G) the minimum degree, ∆(G) the maximum degree, α(G) the independence number, ω(G) the clique number and χ(G) the chromatic number. A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. -
Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs
Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs Erik D. Demaine∗ MohammadTaghi Hajiaghayiy Ken-ichi Kawarabayashiz Abstract between the pieces. For example, many optimization prob- We prove two structural decomposition theorems about graphs ex- lems can be solved exactly on graphs of bounded treewidth; cluding a fixed odd minor H, and show how these theorems can what graphs can be partitioned into a small number k of be used to obtain approximation algorithms for several algorithmic bounded-treewidth pieces? In many cases, each piece gives problems in such graphs. Our decomposition results provide new a lower/upper bound on the optimal solution for the entire structural insights into odd-H-minor-free graphs, on the one hand graph, so solving the problem exactly in each piece gives a generalizing the central structural result from Graph Minor The- k-approximation to the problem. Such a decomposition into ory, and on the other hand providing an algorithmic decomposition bounded-treewidth graphs would also be practical, as many into two bounded-treewidth graphs, generalizing a similar result for NP-hard optimization problems are now solved in practice minors. As one example of how these structural results conquer dif- using dynamic programming on low-treewidth graphs; see, ficult problems, we obtain a polynomial-time 2-approximation for e.g., [Bod05, Ami01, Tho98]. Recently, this decomposi- vertex coloring in odd-H-minor-free graphs, improving on the pre- tion approach has been successfully used for graph coloring, 1−" vious O(jV (H)j)-approximation for such graphs and generalizing which is inapproximable within n for any " > 0 unless the previous 2-approximation for H-minor-free graphs. -
Fractional Colouring and Hadwiger's Conjecture
Journal of Combinatorial Theory, Series B 74, 147152 (1998) Article No. TB981835 Fractional Colouring and Hadwiger's Conjecture Bruce Reed Equipe Combinatoire, Case 189, Universite de Paris VI, 4 Place Jussieu, 75252 Paris Cedex 05, France and Paul Seymour* Department of Mathematics, Princeton University, Princeton, New Jersey 08544 View metadata, citation and similar papersReceived at Octobercore.ac.uk 28, 1997 brought to you by CORE provided by Elsevier - Publisher Connector Let G be a loopless graph with no Kp+1 minor. We prove that the ``fractional chromatic number'' of G is at most 2p; that is, it is possible to assign a rational q(S)0 to every stable set SV(G) so that S % v q(S)=1 for every vertex v, and S q(S)2p. 1998 Academic Press 1. INTRODUCTION In this paper, all graphs are finite and without loops or multiple edges; Kp+1 denotes the complete graph with p+1 vertices; and H is a minor of G if H can be obtained from a subgraph of G by contracting edges. A famous conjecture of Hadwiger [4], dating from 1943, asserts (1.1) (Conjecture) For all p1, every graph with no Kp+1 minor is p-colorable. This is easy for p3, true but difficult for p=4 and 5 (in both cases [9, 6] it is equivalent to the four-colour theorem [1, 2, 7]), and open for p6. It is therefore of interest to examine weakenings of (1.1). If G has no Kp+1 minor, the best upper bound known on its chromatic number is O( p(log p)1Â2). -
On the Odd-Minor Variant of Hadwiger's Conjecture
On the odd-minor variant of Hadwiger’s conjecture Jim Geelen∗, Bert Gerards†, Bruce Reed‡, Paul Seymour§ Adrian Vetta¶ ???; revised April 18, 2011 ∗Department of Combinatorics and Optimization, University of Waterloo. Email: [email protected] †Centrum voor Wiskunde en Informatica, and Eindhoven University of Technology. Email: [email protected] ‡School of Computer Science, McGill University. Email: [email protected] §Department of Mathematics, Princeton University. Email: [email protected] ¶Department of Mathematics and Statistics, and School of Computer Science, McGill University. Email: [email protected] Abstract A Kl-expansion consists of l vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be two-coloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every l, if a graph contains no odd Kl-expansion then its chromatic number is O(l√log l). In doing so, we obtain a characterization of graphs which contain no odd Kl-expansion which is of independent interest. We also prove that given a graph and a subset S of its vertex set, either there are k vertex-disjoint odd paths with endpoints in S, or there is a set X of at most 2k 2 vertices such that every odd path − with both ends in S contains a vertex in X. Finally, we discuss the algorithmic implications of these results. 1 The Main Results All graphs in this paper are finite, and have no loops or parallel edges. -
The Order of the Largest Complete Minor in a Random Graph
THE ORDER OF THE LARGEST COMPLETE MINOR IN A RANDOM GRAPH NIKOLAOS FOUNTOULAKIS, DANIELA KUHN¨ AND DERYK OSTHUS Abstract. Let ccl(G) denote the order of the largest complete minor in a graph G (also called the contraction clique number) and let Gn,p denote a random graph on n vertices with edge probability p. Bollob´as,Catlin and Erd˝os[5] asymptotically determined ccl(Gn,p) when p is a constant.Luczak, Pittel and Wierman [10] gave bounds on ccl(Gn,p) when p is very close to 1/n, i.e. inside the phase transition. We show that for every ε > 0 there exists a constant C suchp that whenever C/n < p < 1 − ε then asymptotically almost surely ccl(G )= (1 ± ε)n/ log (np), where b := 1/(1 − p). If p = C/n for a constant C > 1, n,p b √ then asymptotically almost surely ccl(Gn,p)= Θ( n). This extends the results in [5] and answers a question of Krivelevich and Sudakov [9]. 1. Introduction 1.1. Main results. A graph H is a minor of G if for every vertex h ∈ H there is a connected subset Bh ⊆ V (G) such that all the Bh are disjoint and G contains an edge between Bh 0 and Bh0 whenever hh is an edge of H. The Bh’s are called the branch sets. We denote by ccl(G) the order of the largest complete minor in G. The study of the largest complete minor contained in a given graph has its origins in Hadwiger’s conjecture which states that if the chromatic number of a graph G is at least k, then G contains Kk minor. -
Clique Minors in Graphs with a Forbidden Subgraph
Clique minors in graphs with a forbidden subgraph Matija Bucić∗ Jacob Foxy Benny Sudakovz Abstract The classical Hadwiger conjecture dating back to 1940’s states that any graph of chromatic number at least r has the clique of order r as a minor. Hadwiger’s conjecture is an example of a well-studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph on n vertices of independence number α(G) at most r. If true Hadwiger’s conjecture would imply the existence of a clique minor of order n/α(G). Results of Kühn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition that G is H-free for some bipartite graph H then one can find a polynomially larger clique minor. This has recently been extended to triangle-free graphs by Dvořák and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graph H, answering a question of Dvořák and Yepremyan. In particular, we show that any Ks-free 1+ 1 graph has a clique minor of order cs(n/α(G)) 10(s−2) , for some constant cs depending only on s. The 1 exponent in this result is tight up to a constant factor in front of the s−2 term. 1 Introduction A graph Γ is said to be a minor of a graph G if for every vertex v of Γ we can choose a connected subgraph Gu of G, such that subgraphs Gu are vertex disjoint and G contains an edge between Gv and Gv0 whenever v and v0 make an edge in Γ. -
Hadwiger's Conjecture for Graphs with Forbidden Holes
SIAM J. DISCRETE MATH. c 2017 Society for Industrial and Applied Mathematics Vol. 31, No. 3, pp. 1572{1580 HADWIGER'S CONJECTURE FOR GRAPHS WITH FORBIDDEN HOLES ∗ y z ZI-XIA SONG AND BRIAN THOMAS Abstract. Given a graph G, the Hadwiger number of G, denoted by h(G), is the largest integer k such that G contains the complete graph Kk as a minor. A hole in G is an induced cycle of length at least four. Hadwiger's conjecture from 1943 states that for every graph G, h(G) ≥ χ(G), where χ(G) denotes the chromatic number of G. In this paper we establish more evidence for Hadwiger's conjecture by showing that if a graph G with independence number α(G) ≥ 3 has no hole of length between 4 and 2α(G) − 1, then h(G) ≥ χ(G). We also prove that if a graph G with independence number α(G) ≥ 2 has no hole of length between 4 and 2α(G), then G contains an odd clique minor of size χ(G), that is, such a graph G satisfies the odd Hadwiger's conjecture. Key words. Hadwiger's conjecture, graph minor, quasi-line graph AMS subject classifications. 05C83, 05C15 DOI. 10.1137/16M1086236 1. Introduction. All graphs considered in this paper are finite and have no loops or parallel edges. We begin with some definitions. Let G be a graph. The complement G of G is the graph with vertex set V (G ), such that two vertices are adjacent in G if and only if they are nonadjacent in G.A clique in G is a set of vertices all pairwise adjacent. -
Hadwiger's Conjecture
Hadwiger’s conjecture Paul Seymour∗ Abstract This is a survey of Hadwiger’s conjecture from 1943, that for all t ≥ 0, every graph either can be t-coloured, or has a subgraph that can be contracted to the com- plete graph on t + 1 vertices. This is a tremendous strengthening of the four-colour theorem, and is probably the most famous open problem in graph theory. 1 Introduction The four-colour conjecture (or theorem as it became in 1976), that every planar graph is 4-colourable, was the central open problem in graph theory for a hundred years; and its proof is still not satisfying, requiring as it does the extensive use of a computer. (Let us call it the 4CT.) We would very much like to know the “real” rea- son the 4CT is true; what exactly is it about planarity that implies that four colours suffice? Its statement is so simple and appealing that the massive case analysis of the computer proof surely cannot be the book proof. So there have been attempts to pare down its hypotheses to a minimum core, in the hope of hitting the essentials; to throw away planarity, and impose some weaker condition that still works, and perhaps works with greater transparency so we can comprehend it. This programme has not yet been successful, but it has given rise to some beautiful problems. Of these, the most far-reaching is Hadwiger’s conjecture. (One notable other at- tempt is Tutte’s 1966 conjecture [78] that every 2-edge-connected graph containing no subdivision of the Petersen graph admits a “nowhere-zero 4-flow”, but that is P. -
The Parameterized Complexity of Editing Graphs for Bounded
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 411 (2010) 3181–3187 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs The parameterized complexity of editing graphs for bounded degeneracy Luke Mathieson ∗ School of Engineering and Computing Sciences, University of Durham, South Road, Durham, DH1 3LE, United Kingdom article info a b s t r a c t Article history: We examine the parameterized complexity of the problem of editing a graph to obtain an Received 16 November 2009 r-degenerate graph. We show that for the editing operations vertex deletion and edge dele- Received in revised form 10 May 2010 tion, both separately and combined, the problem is W TPU-complete, and remains W TPU- Accepted 17 May 2010 complete even if the input graph is already .r C 1/-degenerate, or has maximum degree Communicated by G. Ausiello 2r C 1 for all r ≥ 2. We also demonstrate fixed-parameter tractability for several Clique based problems Keywords: when the input graph has bounded degeneracy. Combinatorial problems Computational complexity ' 2010 Elsevier B.V. All rights reserved. Parameterized complexity Degenerate graphs Graph editing 1. Introduction A graph G is r-degenerate if every non-empty subgraph of G has a vertex of degree at most r [8]. The degeneracy of a graph G is the smallest number r such that G is r-degenerate. When we speak of ``degenerate graphs'' we implicitly assume a fixed upper bound r on their degeneracy.