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}
{"title":"Order polytopes of dimension ≤13 are Ehrhart positive","authors":"Feihu Liu , Guoce Xin , Zihao Zhang","doi":"10.1016/j.disc.2025.114833","DOIUrl":"10.1016/j.disc.2025.114833","url":null,"abstract":"<div><div>The order polytopes arising from the finite poset were first introduced and studied by Stanley. For any positive integer <span><math><mi>d</mi><mo>≥</mo><mn>14</mn></math></span>, Liu and Tsuchiya proved that there exists a non-Ehrhart positive order polytope of dimension <em>d</em>. They also proved that any order polytope of dimension <span><math><mi>d</mi><mo>≤</mo><mn>11</mn></math></span> is Ehrhart positive. We confirm that any order polytope of dimension 12 or 13 is Ehrhart positive. This solves an open problem proposed by Liu and Tsuchiya. Besides, we also verify that any <span><math><msup><mrow><mi>h</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-polynomial of order polytope of dimension <span><math><mi>d</mi><mo>≤</mo><mn>13</mn></math></span> is real-rooted.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114833"},"PeriodicalIF":0.7,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268584","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":"Block-transitive t-(k2,k,λ) designs and finite simple exceptional groups of Lie type","authors":"Xingyu Chen , Haiyan Guan","doi":"10.1016/j.disc.2025.114838","DOIUrl":"10.1016/j.disc.2025.114838","url":null,"abstract":"<div><div>Let <em>G</em> be an automorphism group of a nontrivial <em>t</em>-<span><math><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> design. In this paper, we prove that if <em>G</em> is block-transitive, then the socle of <em>G</em> cannot be a finite simple exceptional group of Lie type.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114838"},"PeriodicalIF":0.7,"publicationDate":"2025-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268585","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":"Path decompositions of Eulerian graphs","authors":"Yanan Chu , Yan Wang","doi":"10.1016/j.disc.2025.114830","DOIUrl":"10.1016/j.disc.2025.114830","url":null,"abstract":"<div><div>Gallai's conjecture asserts that every connected graph on <em>n</em> vertices can be decomposed into <span><math><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> paths. For general graphs (possibly disconnected), it was proved that every graph on <em>n</em> vertices can be decomposed into <span><math><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> paths. This is also best possible (consider the graphs consisting of vertex-disjoint triangles). Lovász showed that every <em>n</em>-vertex graph with at most one vertex of even degree can be decomposed into <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> paths. However, Gallai's conjecture is difficult for graphs with many vertices of even degrees. Favaron and Kouider verified Gallai's conjecture for all Eulerian graphs with maximum degree at most 4. In this paper, we show if <em>G</em> is an Eulerian graph on <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span> vertices and the distance between any two triangles in <em>G</em> is at least 3, then <em>G</em> can be decomposed into at most <span><math><mfrac><mrow><mn>3</mn><mi>n</mi></mrow><mrow><mn>5</mn></mrow></mfrac></math></span> paths.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114830"},"PeriodicalIF":0.7,"publicationDate":"2025-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219124","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":"Applications of sparse hypergraph colorings","authors":"Felix Christian Clemen","doi":"10.1016/j.disc.2025.114822","DOIUrl":"10.1016/j.disc.2025.114822","url":null,"abstract":"<div><div>Many problems in extremal combinatorics can be reduced to determining the independence number of a specific auxiliary hypergraph. We present two such problems, one from discrete geometry and one from hypergraph Turán theory. Using results on hypergraph colorings by Cooper-Mubayi and Li-Postle, we demonstrate that for those two problems the trivial lower bound on the independence number can be improved upon:<ul><li><span>•</span><span><div>Erdős, Graham, Ruzsa and Taylor asked to determine the largest size, denoted by <span><math><mi>g</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, of a subset <em>P</em> of the grid <span><math><msup><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> such that every pair of points in <em>P</em> span a different slope. Improving on a lower bound by Zhang from 1993, we show that<span><span><span><math><mi>g</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>Ω</mi><mrow><mo>(</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><msup><mrow><mi>log</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo></mo><mi>n</mi></mrow></mfrac><mo>)</mo></mrow><mo>.</mo></math></span></span></span></div></span></li><li><span>•</span><span><div>Let <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math></span> denote an <em>r</em>-graph with <span><math><mi>r</mi><mo>+</mo><mn>1</mn></math></span> vertices and 3 edges. Recently, Sidorenko proved the following lower bounds for the Turán density of this <em>r</em>-graph: <span><math><mi>π</mi><mo>(</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>)</mo><mo>≥</mo><msup><mrow><mi>r</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></math></span> for every <em>r</em>, and <span><math><mi>π</mi><mo>(</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>)</mo><mo>≥</mo><mo>(</mo><mn>1.7215</mn><mo>−</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msup><mrow><mi>r</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></math></span>. We present an improved asymptotic bound:<span><span><span><math><mi>π</mi><mo>(</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>)</mo><mo>=</mo><mi>Ω</mi><mrow><mo>(</mo><msup><mrow><mi>r</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo></mo><mi>r</mi><mo>)</mo></mrow><mo>.</mo></math></span></span></span></div></span></li></ul></div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114822"},"PeriodicalIF":0.7,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219122","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 new result on disjoint cycles in graphs","authors":"Jie Zhang , Jin Yan","doi":"10.1016/j.disc.2025.114823","DOIUrl":"10.1016/j.disc.2025.114823","url":null,"abstract":"<div><div>Let <em>k</em> be a positive integer. For a graph <em>G</em>, we define ‘<span><math><msubsup><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>’ the minimum value of max<span><math><mo>{</mo><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>d</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>}</mo></math></span> for any pair of nonadjacent vertices <em>x</em> and <em>y</em>. In 1963, Corrádi and Hajnal proved a classical result: every graph <em>G</em> of order at least 3<em>k</em> with minimum degree at least 2<em>k</em> contains <em>k</em> disjoint cycles. Kierstead, Kostochka and Yeager refined Corrádi-Hajnal Theorem by considering the minimum degree of <span><math><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn></math></span>. In this paper, we characterize all graphs on at least <span><math><mn>4</mn><mi>k</mi><mo>+</mo><mn>2</mn></math></span> vertices with <span><math><msubsup><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn></math></span> without <em>k</em> disjoint cycles.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114823"},"PeriodicalIF":0.7,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145227089","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":"Preorder induced by rainbow forbidden subgraphs","authors":"Shun-ichi Maezawa , Akira Saito","doi":"10.1016/j.disc.2025.114817","DOIUrl":"10.1016/j.disc.2025.114817","url":null,"abstract":"<div><div>A subgraph <em>H</em> of an edge-colored graph <em>G</em> is rainbow if all the edges of <em>H</em> receive different colors. If <em>G</em> does not contain a rainbow subgraph isomorphic to <em>H</em>, we say that <em>G</em> is rainbow <em>H</em>-free. For connected graphs <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, if every rainbow <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-free edge-colored complete graph colored in sufficiently many colors is rainbow <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-free, we write <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. The binary relation ≤ is reflexive and transitive, and hence it is a preorder. If <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is a subgraph of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, then trivially <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> holds. On the other hand, there exists a pair <span><math><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> such that <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is a proper supergraph of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> holds. Cui et al. (2021) <span><span>[4]</span></span> characterized these pairs. In this paper, we investigate the pairs <span><math><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> with <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> when neither <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> nor <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is a subgraph of the other.</div><div>While the result by Cui et al. has revealed that the nontrivial pairs <span><math><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> with <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114817"},"PeriodicalIF":0.7,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145227087","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 note on two cycles of consecutive even lengths in graphs","authors":"Binlong Li , Yufeng Pan , Lingjuan Shi","doi":"10.1016/j.disc.2025.114820","DOIUrl":"10.1016/j.disc.2025.114820","url":null,"abstract":"<div><div>Bondy and Vince proved that a graph of minimum degree at least three contains two cycles whose lengths differ by one or two, which was conjectured by Erdős. Gao, Li, Ma and Xie gave an average degree counterpart of Bondy-Vince's result, stating that every <em>n</em>-vertex graph with at least <span><math><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> edges contains two cycles of consecutive even lengths, unless <span><math><mn>4</mn><mo>|</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and every block of <em>G</em> is a clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>. This confirms the case <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span> of Verstraëte's conjecture, which states that every <em>n</em>-vertex graph without <em>k</em> cycles of consecutive even lengths has edge number <span><math><mi>e</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>, with equality if and only if every block of <em>G</em> is a clique of order <span><math><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. Sudakov and Verstraëte further conjectured that if <em>G</em> is a graph with maximum number of edges that does not contain <em>k</em> cycles of consecutive even lengths, then every block of <em>G</em> is a clique of order at most <span><math><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. In this paper, we prove the case <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span> for Sudakov-Verstraëte's conjecture, by extending the results of Gao, Li, Ma and Xie.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114820"},"PeriodicalIF":0.7,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145227086","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 diameter bound for 4-edge-connected C4-free graphs","authors":"Na Chen, Hongxi Liu","doi":"10.1016/j.disc.2025.114821","DOIUrl":"10.1016/j.disc.2025.114821","url":null,"abstract":"<div><div>A graph is called <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free if it does not contain a cycle of length four as a subgraph. Let <em>G</em> be a 4-edge-connected <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free graph of order <em>n</em>. In 2023, Hiebeler, Pardey and Rautenbach proved that the diameter of <em>G</em> is at most <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>3</mn></math></span>, and they conjectured that the factor 1/3 could be improved to 2/7. However, we show that this conjecture does not hold. In this paper, we construct a family of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free, 4-edge-connected graphs of order <em>n</em> with diameter <span><math><mo>(</mo><mn>3</mn><mi>n</mi><mo>−</mo><mn>28</mn><mo>)</mo><mo>/</mo><mn>10</mn></math></span>, and establish a upper bound <span><math><mo>(</mo><mn>6</mn><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo><mo>/</mo><mn>20</mn></math></span> for the diameter, which is optimal up to a small additive constant.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114821"},"PeriodicalIF":0.7,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219123","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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