Discrete Mathematics最新文献

{"title":"Q-independence and the construction of Bh-sets of integers and lattice points","authors":"Melvyn B. Nathanson","doi":"10.1016/j.disc.2025.114726","DOIUrl":"10.1016/j.disc.2025.114726","url":null,"abstract":"<div><div>This paper gives a new <strong>Q</strong>-vector space construction of finite <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span>-sets of integers and lattice points.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114726"},"PeriodicalIF":0.7,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144863575","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":"Gallai's conjecture and the path number of odd semi-cliques","authors":"Yanan Chu ,&nbsp;Genghua Fan ,&nbsp;Chuixiang Zhou","doi":"10.1016/j.disc.2025.114725","DOIUrl":"10.1016/j.disc.2025.114725","url":null,"abstract":"<div><div>Let <em>G</em> be a graph with <em>n</em> vertices. A path decomposition of <em>G</em> is a set of edge-disjoint paths including all the edges of <em>G</em>. Let <span><math><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denote the minimum number of paths in a path decomposition of <em>G</em>. Gallai's Conjecture asserts that if <em>G</em> is connected, then <span><math><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span>. The <em>E</em>-subgraph of <em>G</em> is the subgraph induced by the vertices of even degree in <em>G</em>. A well-known result of Lovász is that if the <em>E</em>-subgraph of <em>G</em> is empty or isomorphic to <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, then <span><math><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></math></span>. In this paper, we prove that if the <em>E</em>-subgraph of <em>G</em> is isomorphic to <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> with <span><math><mi>m</mi><mo>≤</mo><mn>15</mn></math></span>, then <span><math><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo><mo>+</mo><mn>1</mn></math></span>, which implies, under the condition, that Gallai's Conjecture holds when <em>n</em> is odd. A simple graph <em>G</em> on <em>n</em> vertices is called a semi-clique if <span><math><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>&gt;</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. By the definition, if <em>G</em> is a semi-clique on <em>n</em> vertices, then <em>n</em> must be odd and <span><math><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span>. As a corollary of our main result, we obtain that if <em>G</em> is a semi-clique on <em>n</em> vertices, then <span><math><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>4</mn><mi>n</mi><mo>+</mo><mn>6</mn></mrow><mrow><mn>7</mn></mrow></mfrac></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114725"},"PeriodicalIF":0.7,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860521","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":"Chromatic and achromatic numbers of unitary addition Cayley graphs","authors":"Keenan Calhoun,&nbsp;Yeşim Demiroğlu Karabulut,&nbsp;Vincent Pigno,&nbsp;Craig Timmons","doi":"10.1016/j.disc.2025.114735","DOIUrl":"10.1016/j.disc.2025.114735","url":null,"abstract":"<div><div>Let <em>R</em> be a ring. The unitary addition Cayley graph of <em>R</em>, denoted <span><math><mi>U</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, is the graph with vertex <em>R</em>, and two distinct vertices <em>x</em> and <em>y</em> are adjacent if and only if <span><math><mi>x</mi><mo>+</mo><mi>y</mi></math></span> is a unit. We determine a formula for the clique number and chromatic number of such graphs when <em>R</em> is a finite commutative ring with an odd number of elements. This includes the special case when <em>R</em> is <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the integers modulo <em>n</em>, where these parameters had been found under the assumption that <em>n</em> is even, or <em>n</em> is a power of an odd prime. Additionally, we study the achromatic number of <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> in the case that <em>n</em> is the product of two primes. We prove that the achromatic number of <span><math><mi>U</mi><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn><mi>q</mi></mrow></msub><mo>)</mo></math></span> is equal to <span><math><mfrac><mrow><mn>3</mn><mi>q</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> when <span><math><mi>q</mi><mo>&gt;</mo><mn>3</mn></math></span> is a prime. We also prove a lower bound that applies when <span><math><mi>n</mi><mo>=</mo><mi>p</mi><mi>q</mi></math></span> where <em>p</em> and <em>q</em> are distinct odd primes.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114735"},"PeriodicalIF":0.7,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144828918","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":"Clocks are e-positive","authors":"L. Chen ,&nbsp;Y.T. He ,&nbsp;David G.L. Wang","doi":"10.1016/j.disc.2025.114723","DOIUrl":"10.1016/j.disc.2025.114723","url":null,"abstract":"<div><div>Along with his confirmation of the <em>e</em>-positivity of all cycle-chord graphs <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mi>a</mi><mi>b</mi><mn>1</mn></mrow></msub></math></span>, the third author conjectured the <em>e</em>-positivity of all theta graphs <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></msub></math></span>. In this paper, we establish the <em>e</em>-positivity of all clock graphs <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mi>a</mi><mi>b</mi><mn>2</mn></mrow></msub></math></span> by using the composition method. The key idea is to investigate the fibers of certain partial reversal transformation on compositions with all parts at least 2.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114723"},"PeriodicalIF":0.7,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144827353","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":"The dichromatic number of digraphs without induced subdigraphs","authors":"Bin Chen ,&nbsp;Xinmin Hou","doi":"10.1016/j.disc.2025.114729","DOIUrl":"10.1016/j.disc.2025.114729","url":null,"abstract":"<div><div>Let <em>D</em> be a digraph. The dichromatic number of <em>D</em> is the smallest number of colors needed to color the vertices of <em>D</em> such that each color class induces a subdigraph without directed cycles. In this paper, we investigate a conjecture proposed by Aboulker, Charbit and Naserasr, which extends the well known Gyárfás-Sumner conjecture to digraphs. Let <span><math><mover><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mo>→</mo></mrow></mover></math></span> and <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mo>→</mo></mrow></mover></math></span> be a directed path and a directed cycle on <em>k</em> vertices, respectively. Denote by <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> the family of all oriented cycles on 3 vertices. We prove that every <span><math><mo>{</mo><mover><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>7</mn></mrow></msub></mrow><mrow><mo>→</mo></mrow></mover><mo>,</mo><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow><mrow><mo>→</mo></mrow></mover><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>}</mo></math></span>-free oriented graph has dichromatic number at most 190. Additionally, we verify that the dichromatic number of any <span><math><mo>{</mo><mover><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>6</mn></mrow></msub></mrow><mrow><mo>→</mo></mrow></mover><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>}</mo></math></span>-free oriented graph is at most 178, improving a result of Aboulker, Aubian, Charbit and Thomassé.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114729"},"PeriodicalIF":0.7,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144829497","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":"Complete bipartite immersion in graphs with independence number two: A simple proof","authors":"Rong Chen,&nbsp;Zijian Deng","doi":"10.1016/j.disc.2025.114737","DOIUrl":"10.1016/j.disc.2025.114737","url":null,"abstract":"<div><div>A conjecture akin to Hadwiger's conjecture posits that every graph <em>G</em> contains an immersion of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub></math></span>. Vergara showed that, for every <em>n</em>-vertex graph <em>G</em> with independence number two, this is equivalent to saying that <em>G</em> contains an immersion of the complete graph on <span><math><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span> vertices. Recently, Botler et al. showed that every <em>n</em>-vertex graph <em>G</em> with <span><math><mi>α</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span> contains every complete bipartite graph on <span><math><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span> vertices as an immersion. In this paper, we give a much simpler proof of this result.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114737"},"PeriodicalIF":0.7,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144830225","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":"Laguerre inequalities for plane partition","authors":"Zhen-Yu Gao,&nbsp;Larry X.W. Wang","doi":"10.1016/j.disc.2025.114714","DOIUrl":"10.1016/j.disc.2025.114714","url":null,"abstract":"<div><div>In this paper, we establish the Laguerre inequalities of order 2 and 3 for the number of plane partition <span><math><mrow><mi>PL</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. We also demonstrate that <span><math><mrow><mi>PL</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span> satisfies the double Turán inequality. Moreover, we give an upper bound for the threshold of the Laguerre inequality of any order <span><math><mi>r</mi><mo>&gt;</mo><mn>3</mn></math></span> for <span><math><mrow><mi>PL</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114714"},"PeriodicalIF":0.7,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144827352","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":"A characterisation of lines in finite Lie incidence geometries of classical type","authors":"Sira Busch ,&nbsp;Hendrik Van Maldeghem","doi":"10.1016/j.disc.2025.114711","DOIUrl":"10.1016/j.disc.2025.114711","url":null,"abstract":"<div><div>We consider any classical Grassmannian geometry Γ; that is, any projective or polar Grassmann space. Suppose every line in Γ contains <span><math><mi>s</mi><mo>+</mo><mn>1</mn></math></span> points. Then we classify all sets of points in Γ of cardinality <span><math><mi>s</mi><mo>+</mo><mn>1</mn></math></span>, with the property, that no object of opposite type in the corresponding building, is opposite every point of the set. It turns out that such sets are either lines, or hyperbolic lines in symplectic residues, or ovoids in large symplectic subquadrangles of rank 2 residues in characteristic 2. This is a far-reaching extension of a famous and fundamental result of Bose &amp; Burton from the 1960s. We describe a new way to classify geometric lines in finite classical geometries and how our results correspond to blocking sets.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114711"},"PeriodicalIF":0.7,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144829496","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":"On the strict Chvátal-condition and nowhere-zero 3-flows","authors":"Na Yang,&nbsp;Jian-Hua Yin","doi":"10.1016/j.disc.2025.114728","DOIUrl":"10.1016/j.disc.2025.114728","url":null,"abstract":"<div><div>Let <em>G</em> be a simple graph on <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span> vertices and <span><math><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> be the degree sequence of <em>G</em> with <span><math><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mo>⋯</mo><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. The classical Chvátal's theorem states that if <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≥</mo><mi>j</mi><mo>+</mo><mn>1</mn></math></span> or <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>j</mi></mrow></msub><mo>≥</mo><mi>n</mi><mo>−</mo><mi>j</mi></math></span> for each <em>j</em> with <span><math><mn>1</mn><mo>≤</mo><mi>j</mi><mo>&lt;</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, then <em>G</em> is hamiltonian, which implies that <em>G</em> has a nowhere-zero 4-flow. Given an integer <em>i</em> with <span><math><mn>2</mn><mo>≤</mo><mi>i</mi><mo>&lt;</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, we say that the graph <em>G</em> satisfies the strict Chvátal-condition on <em>i</em> if <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≥</mo><mi>j</mi><mo>+</mo><mn>1</mn></math></span> for each <span><math><mi>j</mi><mo>≠</mo><mi>i</mi></math></span> with <span><math><mn>1</mn><mo>≤</mo><mi>j</mi><mo>&lt;</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>i</mi></math></span> and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msub><mo>≥</mo><mi>n</mi><mo>−</mo><mi>i</mi></math></span>. In this paper, we show that if <em>G</em> satisfies the strict Chvátal-condition on <em>i</em> for some <em>i</em> with <span><math><mn>2</mn><mo>≤</mo><mi>i</mi><mo>&lt;</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, then <em>G</em> has no nowhere-zero 3-flow if and only if <span><math><mi>i</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>G</mi><mo>∈</mo><mo>{</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>9</mn></mrow></msub><mo>}</mo></math></span> as described in Fig. 2.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114728"},"PeriodicalIF":0.7,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144826476","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":"Boundedness for proper conflict-free and odd colorings","authors":"A. Jiménez ,&nbsp;K. Knauer ,&nbsp;C.N. Lintzmayer ,&nbsp;M. Matamala ,&nbsp;J.P. Peña ,&nbsp;D.A. Quiroz ,&nbsp;M. Sambinelli ,&nbsp;Y. Wakabayashi ,&nbsp;W. Yu ,&nbsp;J. Zamora","doi":"10.1016/j.disc.2025.114730","DOIUrl":"10.1016/j.disc.2025.114730","url":null,"abstract":"<div><div>The <em>proper conflict-free chromatic number</em>, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, of a graph <em>G</em> is the least positive integer <em>k</em> such that <em>G</em> has a proper <em>k</em>-coloring in which for each non-isolated vertex there is a color appearing exactly once among its neighbors. The <em>proper odd chromatic number</em>, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, of <em>G</em> is the least positive integer <em>k</em> such that <em>G</em> has a proper coloring in which for every non-isolated vertex there is a color appearing an odd number of times among its neighbors. We clearly have <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. We say that a graph class <span><math><mi>G</mi></math></span> is <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow></msub></math></span><em>-bounded</em> (<span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub></math></span><em>-bounded</em>) if there is a function <em>f</em> such that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>f</mi><mo>(</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> (<span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>f</mi><mo>(</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span>) for every <span><math><mi>G</mi><mo>∈</mo><mi>G</mi></math></span>. Caro, Petruševski, and Škrekovski (2023) asked for classes that are linearly <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow></msub></math></span>-bounded (<span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub></math></span>-bounded) and, as a starting point, they showed that every claw-free graph <em>G</em> satisfies <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span>, which implies <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>4</mn><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span>.</div><div>In this paper, we improve the bound for claw-free graphs to a nearly tight bound by showing that such a graph <em>G</em> satisfies <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>6</mn></math></span>, and even <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></m","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114730"},"PeriodicalIF":0.7,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144826477","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}