{"title":"Transversals in a collection of stars or generic trees","authors":"Ethan Y.H. Li , Luyi Li , Ping Li","doi":"10.1016/j.disc.2025.114836","DOIUrl":"10.1016/j.disc.2025.114836","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi></math></span> be a fixed collection of graphs on vertex set <em>V</em> and let <span><math><mi>G</mi></math></span> be a collection of elements in <span><math><mi>F</mi></math></span>. We investigate the transversal problem of finding the maximum value of <span><math><mo>|</mo><mi>G</mi><mo>|</mo></math></span> when <span><math><mi>G</mi></math></span> contains no rainbow element in <span><math><mi>F</mi></math></span>. In this paper, we determine the exact values and characterize all the extremal cases of <span><math><mi>G</mi></math></span> when <span><math><mi>F</mi></math></span> is a collection of stars or generic trees with the same order, respectively.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114836"},"PeriodicalIF":0.7,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268589","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 nucleus of the Johnson graph J(N,D)","authors":"Kazumasa Nomura , Paul Terwilliger","doi":"10.1016/j.disc.2025.114844","DOIUrl":"10.1016/j.disc.2025.114844","url":null,"abstract":"<div><div>This paper is about the nucleus of the Johnson graph <span><math><mi>Γ</mi><mo>=</mo><mi>J</mi><mo>(</mo><mi>N</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> with <span><math><mi>N</mi><mo>></mo><mn>2</mn><mi>D</mi></math></span>. The nucleus is described as follows. Let <em>X</em> denote the vertex set of Γ. Let <span><math><mi>A</mi><mo>∈</mo><msub><mrow><mi>Mat</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> denote the adjacency matrix of Γ. Let <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>D</mi></mrow></msubsup></math></span> denote the <em>Q</em>-polynomial ordering of the primitive idempotents of <em>A</em>. Fix <span><math><mi>x</mi><mo>∈</mo><mi>X</mi></math></span>. The corresponding dual adjacency matrix <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is the diagonal matrix in <span><math><msub><mrow><mi>Mat</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> such that for <span><math><mi>y</mi><mo>∈</mo><mi>X</mi></math></span> the <span><math><mo>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>-entry of <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is equal to the <span><math><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>-entry of <span><math><mo>|</mo><mi>X</mi><mo>|</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. For <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>D</mi></math></span> the diagonal matrix <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>∈</mo><msub><mrow><mi>Mat</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> is the projection onto the <em>i</em>th subconstituent of Γ with respect to <em>x</em>. The matrices <span><math><msubsup><mrow><mo>{</mo><msubsup><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>D</mi></mrow></msubsup></math></span> are the primitive idempotents of <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. The subalgebra <em>T</em> of <span><math><msub><mrow><mi>Mat</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> generated by <em>A</em>, <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is called the subconstituent algebra of Γ with respect to <em>x</em>. Let <span><math><mi>V</mi><mo>=</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msup></math></span> denote the standard module of Γ. For <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>D</mi></math></span> define<span><span><span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow><","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114844"},"PeriodicalIF":0.7,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268083","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 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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
