{"title":"On graphs with maximum difference between game chromatic number and chromatic number","authors":"Lawrence Hollom","doi":"10.1016/j.disc.2024.114271","DOIUrl":null,"url":null,"abstract":"<div><div>In the vertex colouring game on a graph <em>G</em>, Maker and Breaker alternately colour vertices of <em>G</em> from a palette of <em>k</em> colours, with no two adjacent vertices allowed the same colour. Maker seeks to colour the whole graph while Breaker seeks to make some vertex impossible to colour. The game chromatic number of <em>G</em>, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>g</mtext></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the minimum number <em>k</em> of colours for which Maker has a winning strategy for the vertex colouring game.</div><div>Matsumoto proved in 2019 that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>g</mtext></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>−</mo><mn>1</mn></math></span>, and conjectured that the only equality cases are some graphs of small order and the Turán graph <span><math><mi>T</mi><mo>(</mo><mn>2</mn><mi>r</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>. We resolve this conjecture in the affirmative by considering a modification of the vertex colouring game wherein Breaker may remove a vertex instead of colouring it.</div><div>Matsumoto further asked whether a similar result could be proved for the vertex marking game, and we provide an example to show that no such nontrivial result can exist.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114271"},"PeriodicalIF":0.7000,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004023","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In the vertex colouring game on a graph G, Maker and Breaker alternately colour vertices of G from a palette of k colours, with no two adjacent vertices allowed the same colour. Maker seeks to colour the whole graph while Breaker seeks to make some vertex impossible to colour. The game chromatic number of G, , is the minimum number k of colours for which Maker has a winning strategy for the vertex colouring game.
Matsumoto proved in 2019 that , and conjectured that the only equality cases are some graphs of small order and the Turán graph . We resolve this conjecture in the affirmative by considering a modification of the vertex colouring game wherein Breaker may remove a vertex instead of colouring it.
Matsumoto further asked whether a similar result could be proved for the vertex marking game, and we provide an example to show that no such nontrivial result can exist.
在图 G 的顶点着色游戏中,制造者和破坏者交替从 k 种颜色中为 G 的顶点着色,相邻两个顶点不能着相同的颜色。制造者希望给整个图着色,而破坏者则希望让某个顶点无法着色。松本在 2019 年证明了χg(G)-χ(G)≤⌊n/2⌋-1,并猜想唯一相等的情况是一些小阶图和图兰图 T(2r,r)。我们考虑了顶点着色博弈的一种修改,即破坏者可以移除一个顶点而不是给它着色,从而肯定地解决了这一猜想。松本进一步询问是否可以为顶点标记博弈证明类似的结果,我们举例说明不可能存在这种非难结果。
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.