{"title":"On the matching problem in random hypergraphs","authors":"Peter Frankl , Jiaxi Nie , Jian Wang","doi":"10.1016/j.disc.2025.114839","DOIUrl":"10.1016/j.disc.2025.114839","url":null,"abstract":"<div><div>We study a variant of the Erdős Matching Problem in random hypergraphs. Let <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> denote the Erdős-Rényi random <em>k</em>-uniform hypergraph on <em>n</em> vertices where each possible edge is included with probability <em>p</em>. We show that when <span><math><mi>n</mi><mo>≫</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>s</mi></math></span> and <em>p</em> is not too small, with high probability, the maximum number of edges in a sub-hypergraph of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> with matching number <em>s</em> is obtained by the trivial sub-hypergraphs, i.e. the sub-hypergraph consisting of all edges containing at least one vertex in a fixed set of <em>s</em> vertices.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114839"},"PeriodicalIF":0.7,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268084","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":"Constant congestion linkages in polynomially strong digraphs in polynomial time","authors":"Raul Lopes , Ignasi Sau","doi":"10.1016/j.disc.2025.114808","DOIUrl":"10.1016/j.disc.2025.114808","url":null,"abstract":"<div><div>Given positive integers <em>k</em> and <em>c</em>, we say that a digraph <em>D</em> is <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></span><em>-linked</em> if for every pair of ordered sets <span><math><mo>{</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span> and <span><math><mo>{</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span> of vertices of <em>D</em>, there are paths <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> such that for <span><math><mi>i</mi><mo>∈</mo><mo>[</mo><mi>k</mi><mo>]</mo></math></span> each <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a path from <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> to <span><math><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and every vertex of <em>D</em> appears in at most <em>c</em> of those paths. A classical result by Thomassen [Combinatorica, 1991] states that, for every fixed <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, there is no integer <em>p</em> such that every <em>p</em>-strong digraph is <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-linked.</div><div>Edwards et al. [ESA, 2017] showed that every digraph <em>D</em> with directed treewidth at least some function <span><math><mi>f</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> contains a large bramble of congestion 2. Then, they showed that every <span><math><mo>(</mo><mn>36</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>)</mo></math></span>-strong digraph containing a bramble of congestion 2 and size roughly <span><math><mn>188</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> is <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-linked. Since the directed treewidth of a digraph has to be at least its strong connectivity, this implies that there is a function <span><math><mi>L</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> such that every <span><math><mi>L</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span>-strong digraph is <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-linked. The result by Edwards et al. was improved by Campos et al. [ESA, 2023], who showed that any <em>k</em>-strong digraph containing a bramble of size at least <span><math><mn>2</mn><mi>k</mi><mo>(</mo><mi>c</mi><mo>⋅</mo><mi>k</mi><mo>−</mo><mi>c</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>+</mo><mi>c</mi><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and congesti","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114808"},"PeriodicalIF":0.7,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265225","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}
Dipayan Chakraborty , Florent Foucaud , Michael A. Henning , Tuomo Lehtilä
{"title":"Identifying codes in graphs of given maximum degree: Characterizing trees","authors":"Dipayan Chakraborty , Florent Foucaud , Michael A. Henning , Tuomo Lehtilä","doi":"10.1016/j.disc.2025.114826","DOIUrl":"10.1016/j.disc.2025.114826","url":null,"abstract":"<div><div>An <em>identifying code</em> of a closed-twin-free graph <em>G</em> is a dominating set <em>S</em> of vertices of <em>G</em> such that any two vertices in <em>G</em> have a distinct intersection between their closed neighborhoods and <em>S</em>. It was conjectured that there exists an absolute constant <em>c</em> such that for every connected graph <em>G</em> of order <em>n</em> and maximum degree Δ, the graph <em>G</em> admits an identifying code of size at most <span><math><mo>(</mo><mfrac><mrow><mi>Δ</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>Δ</mi></mrow></mfrac><mo>)</mo><mi>n</mi><mo>+</mo><mi>c</mi></math></span>. We provide significant support for this conjecture by exactly characterizing every tree requiring a positive constant <em>c</em> together with the exact value of the constant. Hence, proving the conjecture for trees. For <span><math><mi>Δ</mi><mo>=</mo><mn>2</mn></math></span> (the graph is a path or a cycle), it is long known that <span><math><mi>c</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math></span> suffices. For trees, for each <span><math><mi>Δ</mi><mo>≥</mo><mn>3</mn></math></span>, we show that <span><math><mi>c</mi><mo>=</mo><mn>1</mn><mo>/</mo><mi>Δ</mi><mo>≤</mo><mn>1</mn><mo>/</mo><mn>3</mn></math></span> suffices and that <em>c</em> is required to have a positive value only for a finite number of trees. In particular, for <span><math><mi>Δ</mi><mo>=</mo><mn>3</mn></math></span>, there are 12 trees with a positive constant <em>c</em> and, for each <span><math><mi>Δ</mi><mo>≥</mo><mn>4</mn></math></span>, the only tree with positive constant <em>c</em> is the Δ-star. Our proof is based on induction and utilizes recent results from Foucaud and Lehtilä (2022) <span><span>[17]</span></span>. We remark that there are infinitely many trees for which the bound is tight when <span><math><mi>Δ</mi><mo>=</mo><mn>3</mn></math></span>; for every <span><math><mi>Δ</mi><mo>≥</mo><mn>4</mn></math></span>, we construct an infinite family of trees of order <em>n</em> with identification number very close to the bound, namely <span><math><mrow><mo>(</mo><mfrac><mrow><mi>Δ</mi><mo>−</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>Δ</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow><mrow><mi>Δ</mi><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>Δ</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mfrac><mo>)</mo></mrow><mi>n</mi><mo>></mo><mo>(</mo><mfrac><mrow><mi>Δ</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>Δ</mi></mrow></mfrac><mo>)</mo><mi>n</mi><mo>−</mo><mfrac><mrow><mi>n</mi></mrow><mrow><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></span>. Furthermore, we also give a new tight upper bound for identification number on trees by showing that the sum of the domination and identification numbers of any tree <em>T</em> is at most its number of vertices.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114826"},"PeriodicalIF":0.7,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265228","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":"Spectral extremal results on edge blow-up of graphs","authors":"Longfei Fang , Huiqiu Lin","doi":"10.1016/j.disc.2025.114835","DOIUrl":"10.1016/j.disc.2025.114835","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> be the maximum size and the maximum spectral radius of an <em>F</em>-free graph of order <em>n</em>, respectively. The value <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> is called the spectral extremal value of <em>F</em>. Nikiforov (2009) <span><span>[24]</span></span> gave the spectral Stability Lemma, which implies that for every <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, sufficiently large <em>n</em> and a non-bipartite graph <em>H</em> with chromatic number <span><math><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>, the extremal graph for <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> can be obtained from the Turán graph <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> by adding and deleting at most <span><math><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> edges. It is still a challenging problem to determine the exact spectral extremal values of many non-bipartite graphs. Given a graph <em>F</em> and an integer <span><math><mi>p</mi><mo>≥</mo><mn>2</mn></math></span>, the edge blow-up of <em>F</em>, denoted by <span><math><msup><mrow><mi>F</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>, is the graph obtained from replacing each edge in <em>F</em> by a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> where the new vertices of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> are all distinct. In this paper, we determine the exact spectral extremal values of the edge blow-up of all non-bipartite graphs and provide the asymptotic spectral extremal values of the edge blow-up of all bipartite graphs for sufficiently large <em>n</em>, which can be seen as a spectral version of the theorem on <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> given by Yuan (2022) <span><span>[34]</span></span>. As applications, on the one hand, we generalize several previous results on <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> for <em>F</em> being a matching and a star. On the other hand, we obtain the exact values of <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo><","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114835"},"PeriodicalIF":0.7,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265223","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":"Strictly critical snarks with girth or cyclic connectivity equal to 6","authors":"Ján Mazák , Jozef Rajník , Martin Škoviera","doi":"10.1016/j.disc.2025.114827","DOIUrl":"10.1016/j.disc.2025.114827","url":null,"abstract":"<div><div>A snark – connected cubic graph with chromatic index 4 – is critical if the graph resulting from the removal of any pair of distinct adjacent vertices is 3-edge-colourable; it is bicritical if the same is true for any pair of distinct vertices. A snark is strictly critical if it is critical but not bicritical. Very little is known about strictly critical snarks. Computational evidence suggests that strictly critical snarks constitute a tiny minority of all critical snarks. Strictly critical snarks of order <em>n</em> exist if and only if <em>n</em> is even and at least 32, and for each such order there is at least one strictly critical snark with cyclic connectivity 4. A sparse infinite family of cyclically 5-connected strictly critical snarks is also known, but those with cyclic connectivity greater than 5 have not been discovered so far. In this paper we fill the gap by constructing cyclically 6-connected strictly critical snarks of each even order <span><math><mi>n</mi><mo>≥</mo><mn>342</mn></math></span>. In addition, we construct cyclically 5-connected strictly critical snarks of girth 6 for every even <span><math><mi>n</mi><mo>≥</mo><mn>66</mn></math></span> with <span><math><mi>n</mi><mo>≡</mo><mn>2</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>8</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114827"},"PeriodicalIF":0.7,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265226","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 arborescence packing augmentation in hypergraphs","authors":"Pierre Hoppenot, Zoltán Szigeti","doi":"10.1016/j.disc.2025.114837","DOIUrl":"10.1016/j.disc.2025.114837","url":null,"abstract":"<div><div>We deepen the link between two classic areas of combinatorial optimization: augmentation and packing arborescences. We consider the following type of questions: What is the minimum number of arcs to be added to a digraph so that in the resulting digraph there exists some special kind of packing of arborescences? We answer this question for two problems: <em>h</em>-regular <span>M</span>-independent-rooted <span><math><mo>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span>-bounded <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span>-limited packing of mixed hyperarborescences and <em>h</em>-regular <span><math><mo>(</mo><mi>ℓ</mi><mo>,</mo><msup><mrow><mi>ℓ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></math></span>-bordered <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span>-limited packing of <em>k</em> hyperbranchings. We also solve the undirected counterpart of the latter, that is the augmentation problem for <em>h</em>-regular <span><math><mo>(</mo><mi>ℓ</mi><mo>,</mo><msup><mrow><mi>ℓ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></math></span>-bordered <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span>-limited packing of <em>k</em> rooted hyperforests. Our results provide a common generalization of a great number of previous results.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114837"},"PeriodicalIF":0.7,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268586","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 nonrepetitive coloring of grids","authors":"Tianyi Tao","doi":"10.1016/j.disc.2025.114828","DOIUrl":"10.1016/j.disc.2025.114828","url":null,"abstract":"<div><div>For a graph <em>G</em>, a vertex coloring <em>f</em> is called nonrepetitive if for all <span><math><mi>k</mi><mo>∈</mo><mi>N</mi></math></span> and all <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub><mo>=</mo><mo>〈</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub><mo>〉</mo></math></span> (path of 2<em>k</em> vertices) in <em>G</em>, there must be some <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></math></span> such that <span><math><mi>f</mi><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>≠</mo><mi>f</mi><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>)</mo></math></span>.</div><div>We use <span><math><mi>π</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> to denote the minimum number of colors required for <em>G</em> to be nonrepetitively colored.</div><div>In 1906, Thue proved that <span><math><mi>π</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>≤</mo><mn>3</mn></math></span> for all <em>n</em>. In this paper, we focus on grids, which are the Cartesian products of paths. We prove that <span><math><mn>5</mn><mo>≤</mo><mi>π</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>□</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>≤</mo><mn>12</mn></math></span> for sufficiently large <em>n</em>, where the previous best lower bound was 4 and upper bound was 16. Moreover, we also discuss nonrepetitive coloring of the Cartesian product of complete graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114828"},"PeriodicalIF":0.7,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265227","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}
Paula M.S. Fialho , Emanuel Juliano , Aldo Procacci
{"title":"On the zero-free region for the chromatic polynomial of graphs with maximum degree Δ and girth g","authors":"Paula M.S. Fialho , Emanuel Juliano , Aldo Procacci","doi":"10.1016/j.disc.2025.114825","DOIUrl":"10.1016/j.disc.2025.114825","url":null,"abstract":"<div><div>The purpose of the present paper is to provide, for all pairs of integers <span><math><mo>(</mo><mi>Δ</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> with <span><math><mi>Δ</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><mi>g</mi><mo>≥</mo><mn>3</mn></math></span>, a positive number <span><math><mi>C</mi><mo>(</mo><mi>Δ</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span> such that chromatic polynomial <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> of a graph <span><math><mi>G</mi></math></span> with maximum degree Δ and finite girth <em>g</em> is free of zero if <span><math><mo>|</mo><mi>q</mi><mo>|</mo><mo>≥</mo><mi>C</mi><mo>(</mo><mi>Δ</mi><mo>,</mo><mi>g</mi><mo>)</mo></math></span>. Our bounds enlarge the zero-free region in the complex plane of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> in comparison to all previous bounds. In particular, for small values of Δ our estimates yield an expressive improvement on the bounds recently obtained by Jenssen, Patel and Regts in [J. Comb. Theor. B, 169 (2024)], while they coincide with their estimates when <span><math><mi>Δ</mi><mo>→</mo><mo>∞</mo></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114825"},"PeriodicalIF":0.7,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268588","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}
Xiaoxue Hu , Jiangxu Kong , Weifan Wang , Wanshun Yang
{"title":"A note on the r-hued coloring of planar graphs","authors":"Xiaoxue Hu , Jiangxu Kong , Weifan Wang , Wanshun Yang","doi":"10.1016/j.disc.2025.114829","DOIUrl":"10.1016/j.disc.2025.114829","url":null,"abstract":"<div><div>Let <span><math><mi>r</mi><mo>≥</mo><mn>1</mn></math></span> be an integer. The <em>r</em>-hued chromatic number <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> is the smallest integer <em>k</em> for which <em>G</em> admits a proper <em>k</em>-coloring for the vertices such that the number of colors used in the neighborhood of every vertex <em>v</em> is at least <span><math><mi>min</mi><mo></mo><mo>{</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>,</mo><mi>r</mi><mo>}</mo></math></span>. Let <em>G</em> be a planar graph and <span><math><mi>r</mi><mo>≥</mo><mn>8</mn></math></span>. In this paper we show that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mi>r</mi><mo>+</mo><mn>8</mn></math></span>, which improves a result by Song and Lai (2018) <span><span>[12]</span></span> that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mi>r</mi><mo>+</mo><mn>16</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114829"},"PeriodicalIF":0.7,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265224","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":"Quantum walks on join graphs","authors":"Steve Kirkland, Hermie Monterde","doi":"10.1016/j.disc.2025.114832","DOIUrl":"10.1016/j.disc.2025.114832","url":null,"abstract":"<div><div>The join <span><math><mi>X</mi><mo>∨</mo><mi>Y</mi></math></span> of two graphs <em>X</em> and <em>Y</em> is the graph obtained by joining each vertex of <em>X</em> to each vertex of <em>Y</em>. We explore the behaviour of a continuous quantum walk on a join graph with positive edge weights having the adjacency matrix or Laplacian matrix as its associated Hamiltonian, where the underlying graphs are assumed to be regular when dealing with the adjacency matrix. We characterize strong cospectrality, periodicity and perfect state transfer (PST) in a join graph. We also determine conditions in which strong cospectrality, periodicity and PST are preserved in the join. Under certain conditions, we show that there are graphs with no PST that exhibit PST when joined with another graph. We also show that <span><math><mo>|</mo><mo>|</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>M</mi></mrow></msub><msub><mrow><mo>(</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></msub><mo>|</mo><mo>−</mo><mo>|</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>M</mi></mrow></msub><msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></msub><mo>|</mo><mo>|</mo><mo>≤</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mo>|</mo><mi>V</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>|</mo></mrow></mfrac></math></span> for all vertices <em>u</em> and <em>v</em> of <em>X</em>, where <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> denote the transition matrices of <span><math><mi>X</mi><mo>∨</mo><mi>Y</mi></math></span> and <em>X</em> respectively relative to the adjacency or Laplacian matrix. We demonstrate that the bound <span><math><mfrac><mrow><mn>2</mn></mrow><mrow><mo>|</mo><mi>V</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>|</mo></mrow></mfrac></math></span> is tight for infinite families of graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114832"},"PeriodicalIF":0.7,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268587","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}
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