{"title":"On weak cop numbers of transitive graphs","authors":"Florian Lehner","doi":"10.1016/j.disc.2025.114559","DOIUrl":"10.1016/j.disc.2025.114559","url":null,"abstract":"<div><div>The weak cop number of infinite graphs can be seen as a coarse-geometric analogue to the cop number of finite graphs. We show that every vertex transitive graph with at least one thick end has infinite weak cop number. It follows that every connected, vertex transitive graph has weak cop number 1 or ∞, answering a question posed by Lee, Martínez-Pedroza, and Rodríguez-Quinche, and reiterated in recent preprints by Appenzeller and Klinge, and by Esperet, Gahlawat, and Giocanti.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114559"},"PeriodicalIF":0.7,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143934869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Distribution of hooks in self-conjugate partitions","authors":"William Craig , Ken Ono , Ajit Singh","doi":"10.1016/j.disc.2025.114563","DOIUrl":"10.1016/j.disc.2025.114563","url":null,"abstract":"<div><div>We confirm the speculation that the distribution of <em>t</em>-hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the number of hooks of length <em>t</em> among the size <em>n</em> self-conjugate partitions is asymptotically normally distributed with mean <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and variance <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span><span><span><span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>∼</mo><mfrac><mrow><msqrt><mrow><mn>6</mn><mi>n</mi></mrow></msqrt></mrow><mrow><mi>π</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn></mrow><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mn>4</mn></mrow></mfrac><mspace></mspace><mspace></mspace><mspace></mspace><mtext>and</mtext><mspace></mspace><mspace></mspace><mspace></mspace><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>(</mo><mi>n</mi><mo>)</mo><mo>∼</mo><mfrac><mrow><mrow><mo>(</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>6</mn><mo>)</mo></mrow><msqrt><mrow><mn>6</mn><mi>n</mi></mrow></msqrt></mrow><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>:</mo><mo>=</mo><mn>1</mn></math></span> if <em>t</em> is odd and is 0 otherwise.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114563"},"PeriodicalIF":0.7,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143934868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sudip Bera , Hiranya Kishore Dey , Kamal Lochan Patra , Binod Kumar Sahoo
{"title":"On the domination number of proper power graphs of finite groups","authors":"Sudip Bera , Hiranya Kishore Dey , Kamal Lochan Patra , Binod Kumar Sahoo","doi":"10.1016/j.disc.2025.114557","DOIUrl":"10.1016/j.disc.2025.114557","url":null,"abstract":"<div><div>The proper power graph <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a finite group <em>G</em> is the simple graph whose vertices are the <em>nonindentity</em> elements of <em>G</em> and two distinct vertices are adjacent if one of them is a power of the other. In this paper, we study the domination number <span><math><mi>γ</mi><mo>(</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> by relating it with the number of distinct prime order subgroups of <em>G</em>. For a nilpotent group <em>G</em>, we give a sharp upper bound for <span><math><mi>γ</mi><mo>(</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span>. When <em>G</em> is a direct product of two nontrivial groups <em>H</em> and <em>K</em>, we give a sharp lower bound for <span><math><mi>γ</mi><mo>(</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> in terms of the number of components of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>H</mi><mo>)</mo></math></span> and <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>K</mi><mo>)</mo></math></span>. As an application, we determine <span><math><mi>γ</mi><mo>(</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> when <em>G</em> is a nilpotent group whose order is divisible by at most two distinct primes.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114557"},"PeriodicalIF":0.7,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143929063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Paintability of r-chromatic graphs","authors":"Peter Bradshaw , Jinghan A. Zeng","doi":"10.1016/j.disc.2025.114558","DOIUrl":"10.1016/j.disc.2025.114558","url":null,"abstract":"<div><div>The online list coloring game is a two-player graph-coloring game played on a graph <em>G</em> as follows. On each turn, a Lister reveals a new color <em>c</em> at some subset <span><math><mi>S</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of uncolored vertices, and then a Painter chooses an independent subset of <em>S</em> to which to assign <em>c</em>. As the game is played, the revealed colors at each vertex <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> form a color set <span><math><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo></math></span>, often called a list. The <em>paintability</em> of <em>G</em> measures the minimum value <em>k</em> for which Painter has a strategy to complete a coloring of <em>G</em> in such a way that <span><math><mo>|</mo><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>|</mo><mo>≤</mo><mi>k</mi></math></span> for each vertex <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. The paintability of a graph is an upper bound for its list chromatic number, or choosability.</div><div>The online list coloring game is a special case of the <em>DP-painting</em> game, which is defined similarly using the setting of DP-coloring. In the DP-painting game, the Lister reveals correspondence covers of a graph <em>G</em> rather than colors, and the Painter chooses independent subsets of these covers. The DP-painting game has a parameter known as <em>DP-paintability</em> which is analogous to paintability.</div><div>In this paper, we consider upper bounds for the paintability and DP-paintability of a graph <em>G</em> with large maximum degree Δ and chromatic number at most some fixed value <em>r</em>. We prove that the paintability of <em>G</em> is at most <span><math><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><mi>r</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow><mi>Δ</mi><mo>+</mo><mn>2</mn></math></span> and that the DP-paintability of <em>G</em> is at most <span><math><mi>Δ</mi><mo>−</mo><mi>Ω</mi><mo>(</mo><msqrt><mrow><mi>Δ</mi><mi>log</mi><mo></mo><mi>Δ</mi></mrow></msqrt><mo>)</mo></math></span>. We prove our first upper bound using Alon-Tarsi orientations, and we prove our second upper bound by considering the <em>strict type-</em>3 <em>degeneracy</em> parameter recently introduced by Zhou, Zhu, and Zhu.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114558"},"PeriodicalIF":0.7,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143929064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An odd 4-coloring of a maximal outerplanar graph","authors":"Masaki Kashima , Shun-ichi Maezawa , Kakeru Osako , Kenta Ozeki , Shoichi Tsuchiya","doi":"10.1016/j.disc.2025.114556","DOIUrl":"10.1016/j.disc.2025.114556","url":null,"abstract":"<div><div>An odd coloring of a graph <em>G</em> is a proper coloring with the following property: For every vertex <em>v</em> of <em>G</em>, there exists a color <em>i</em> such that there are an odd number of vertices of color <em>i</em> in the neighborhood of <em>v</em>. Caro, Petruševski, and Škrekovski proved that every outerplanar graph admits an odd 5-coloring. Since the cycle of length 5 does not admit an odd 4-coloring, this result is best possible. In this paper, we prove that every maximal outerplanar graph admits an odd 4-coloring. We also show that the list version holds.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114556"},"PeriodicalIF":0.7,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An Erdős–Ko–Rado type theorem for subgraphs of perfect matchings","authors":"Dániel T. Nagy","doi":"10.1016/j.disc.2025.114560","DOIUrl":"10.1016/j.disc.2025.114560","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> be a 2<em>n</em>-vertex graph with <em>n</em> pairwise disjoint edges and let <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the family of subsets of <span><math><mi>V</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> that span exactly <em>p</em> edges and <em>s</em> isolated vertices. We prove that for <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>p</mi><mo>+</mo><mi>s</mi></math></span> this family has the Erdős–Ko–Rado property: the size of the largest intersecting family is equal to the number of sets containing a fixed vertex. The bound <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>p</mi><mo>+</mo><mi>s</mi></math></span> is the best possible, improving a recent theorem with <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>p</mi><mo>+</mo><mn>2</mn><mi>s</mi></math></span> by Fuentes and Kamat.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114560"},"PeriodicalIF":0.7,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143917513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Line graphs with the largest eigenvalue multiplicity","authors":"Wenhao Zhen, Dein Wong , Songnian Xu","doi":"10.1016/j.disc.2025.114562","DOIUrl":"10.1016/j.disc.2025.114562","url":null,"abstract":"<div><div>For a connected graph <em>G</em>, we denote by <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>λ</mi><mo>)</mo></math></span>, <span><math><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> the line graph of <em>G</em>, the eigenvalue multiplicity of <em>λ</em> in <em>G</em>, the cyclomatic number and the number of pendant vertices in <em>G</em>, respectively. In 2023, Yang et al. <span><span>[12]</span></span> proved that <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>p</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for any tree <em>T</em> with <span><math><mi>p</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>≥</mo><mn>3</mn></math></span>, and characterized all trees <em>T</em> with <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>λ</mi><mo>)</mo><mo>=</mo><mi>p</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>. In 2024, Chang et al. <span><span>[2]</span></span> proved that, if <em>G</em> is not a cycle, then <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, and they characterized all graphs <em>G</em> with <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>2</mn><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>. The authors of <span><span>[2]</span></span> particularly stated that it seems somewhat difficult to characterize the extremal graphs <em>G</em> with <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>λ</mi><mo>)</mo><mo>=</mo><mn>2</mn><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for an arbitrary eigenvalue <em>λ</em> of <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In this paper, we give this problem a complete solution.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114562"},"PeriodicalIF":0.7,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Meysam Miralaei , Ali Mohammadian , Behruz Tayfeh-Rezaie , Maksim Zhukovskii
{"title":"Saturation numbers of bipartite graphs in random graphs","authors":"Meysam Miralaei , Ali Mohammadian , Behruz Tayfeh-Rezaie , Maksim Zhukovskii","doi":"10.1016/j.disc.2025.114561","DOIUrl":"10.1016/j.disc.2025.114561","url":null,"abstract":"<div><div>For a given graph <em>F</em>, the <em>F</em>-saturation number of a graph <em>G</em>, denoted by <span><math><mrow><mi>sat</mi></mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, is the minimum number of edges in an edge-maximal <em>F</em>-free subgraph of <em>G</em>. In 2017, Korándi and Sudakov determined <figure><img></figure> asymptotically, where <figure><img></figure> denotes the Erdős–Rényi random graph and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> is the complete graph on <em>r</em> vertices. In this paper, among other results, we present an asymptotic upper bound on <figure><img></figure> for any bipartite graph <em>F</em> and also an asymptotic lower bound on <figure><img></figure> for any complete bipartite graph <em>F</em>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114561"},"PeriodicalIF":0.7,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143904231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Group irregularity strength of disconnected graphs","authors":"Sylwia Cichacz, Barbara Krupińska","doi":"10.1016/j.disc.2025.114548","DOIUrl":"10.1016/j.disc.2025.114548","url":null,"abstract":"<div><div>We investigate the <em>group irregular strength</em> <span><math><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> of graphs, i.e. the smallest value of <em>s</em> such that for any Abelian group Γ of order <em>s</em> exists a function <span><math><mi>g</mi><mo>:</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mi>Γ</mi></math></span> such that sums of edge labels at every vertex is distinct. We give results for bound and exact values of <span><math><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> for graphs without small stars as components.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114548"},"PeriodicalIF":0.7,"publicationDate":"2025-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143902114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Optimal coloring of (P2 + P3, gem)-free graphs","authors":"Arnab Char, T. Karthick","doi":"10.1016/j.disc.2025.114554","DOIUrl":"10.1016/j.disc.2025.114554","url":null,"abstract":"<div><div>Given a graph <em>G</em>, the parameters <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> respectively denote the chromatic number and the clique number of <em>G</em>. A function <span><math><mi>f</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>N</mi></math></span> such that <span><math><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>1</mn></math></span> and <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>≥</mo><mi>x</mi></math></span>, for all <span><math><mi>x</mi><mo>∈</mo><mi>N</mi></math></span> is called a <em>χ-binding function</em> for the given class of graphs <span><math><mi>G</mi></math></span> if every <span><math><mi>G</mi><mo>∈</mo><mi>G</mi></math></span> satisfies <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>f</mi><mo>(</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span>, and the <em>smallest χ-binding function</em> <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> for <span><math><mi>G</mi></math></span> is defined as <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>max</mi><mo></mo><mo>{</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mi>G</mi><mo>∈</mo><mi>G</mi><mtext> and </mtext><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>x</mi><mo>}</mo></math></span>. In general, the problem of obtaining the smallest <em>χ</em>-binding function for the given class of graphs seems to be extremely hard, and only a few classes of graphs are studied in this direction. In this paper, we study the class of (<span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, gem)-free graphs, and prove that the function <span><math><mi>ϕ</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>N</mi></math></span> defined by <span><math><mi>ϕ</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>1</mn></math></span>, <span><math><mi>ϕ</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>=</mo><mn>4</mn></math></span>, <span><math><mi>ϕ</mi><mo>(</mo><mn>3</mn><mo>)</mo><mo>=</mo><mn>6</mn></math></span> and <span><math><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mrow><mo>⌈</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>(</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>⌉</mo></mrow></math></span>, for <span><math><mi>x</mi><mo>≥</mo><mn>4</mn></math></span> is the smallest <em>χ</em>-binding function for the class of (<span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, gem)-free graphs. Also we completely characterize the class of (<span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, gem)-free","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114554"},"PeriodicalIF":0.7,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143900018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
