Simona Boyadzhiyska , Shagnik Das , Thomas Lesgourgues , Kalina Petrova
{"title":"Odd-Ramsey numbers of complete bipartite graphs","authors":"Simona Boyadzhiyska , Shagnik Das , Thomas Lesgourgues , Kalina Petrova","doi":"10.1016/j.ejc.2025.104235","DOIUrl":"10.1016/j.ejc.2025.104235","url":null,"abstract":"<div><div>In his study of graph codes, Alon introduced the concept of the <em>odd-Ramsey</em> number of a family of graphs <span><math><mi>H</mi></math></span> in <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, defined as the minimum number of colours needed to colour the edges of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> so that every copy of a graph <span><math><mrow><mi>H</mi><mo>∈</mo><mi>H</mi></mrow></math></span> intersects some colour class in an odd number of edges. In this paper, we focus on complete bipartite graphs. First, we completely resolve the problem when <span><math><mi>H</mi></math></span> is the family of all spanning complete bipartite graphs on <span><math><mi>n</mi></math></span> vertices. We then focus on its subfamilies, that is, <span><math><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>t</mi></mrow></msub><mo>:</mo><mi>t</mi><mo>∈</mo><mi>T</mi><mo>}</mo></mrow></math></span> for a fixed set of integers <span><math><mrow><mi>T</mi><mo>⊆</mo><mrow><mo>[</mo><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow><mo>]</mo></mrow></mrow></math></span>. We prove that the odd-Ramsey problem is equivalent to determining the maximum dimension of a linear binary code avoiding codewords of given weights, and leverage known results from coding theory to deduce asymptotically tight bounds in our setting. We conclude with bounds for the odd-Ramsey numbers of fixed (that is, non-spanning) complete bipartite subgraphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104235"},"PeriodicalIF":0.9,"publicationDate":"2025-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049185","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":"Homotopy types of Hom complexes of graph homomorphisms whose codomains are square-free","authors":"Soichiro Fujii , Kei Kimura , Yuta Nozaki","doi":"10.1016/j.ejc.2025.104238","DOIUrl":"10.1016/j.ejc.2025.104238","url":null,"abstract":"<div><div>Given finite simple graphs <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>, the Hom complex <span><math><mrow><mi>Hom</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is a polyhedral complex having the graph homomorphisms <span><math><mrow><mi>G</mi><mo>→</mo><mi>H</mi></mrow></math></span> as the vertices. We determine the homotopy type of each connected component of <span><math><mrow><mi>Hom</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mi>H</mi></math></span> is square-free, meaning that it does not contain the 4-cycle graph <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> as a subgraph. Specifically, for a connected <span><math><mi>G</mi></math></span> and a square-free <span><math><mi>H</mi></math></span>, we show that each connected component of <span><math><mrow><mi>Hom</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is homotopy equivalent to a wedge sum of circles. We further show that, given any graph homomorphism <span><math><mrow><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>H</mi></mrow></math></span> to a square-free <span><math><mi>H</mi></math></span>, one can determine the homotopy type of the connected component of <span><math><mrow><mi>Hom</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> containing <span><math><mi>f</mi></math></span> algorithmically.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104238"},"PeriodicalIF":0.9,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049184","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}
Borut Lužar , Edita Máčajová , Roman Soták , Diana Švecová
{"title":"List strong and list normal edge-coloring of (sub)cubic graphs","authors":"Borut Lužar , Edita Máčajová , Roman Soták , Diana Švecová","doi":"10.1016/j.ejc.2025.104243","DOIUrl":"10.1016/j.ejc.2025.104243","url":null,"abstract":"<div><div>A <em>strong edge-coloring</em> of a graph is a proper edge-coloring, in which the edges of every path of length 3 receive distinct colors; in other words, every pair of edges at distance at most 2 must be colored differently. The least number of colors needed for a strong edge-coloring of a graph is the <em>strong chromatic index</em>. We consider the list version of the coloring and prove that the list strong chromatic index of graphs with maximum degree 3 is at most 10. This bound is tight and improves the previous bound of 11 colors.</div><div>We also consider the question whether the strong chromatic index and the list strong chromatic index always coincide. We answer it in negative by presenting an infinite family of graphs for which the two invariants differ. For the special case of the Petersen graph, we show that its list strong chromatic index equals 7, while its strong chromatic index is 5. Up to our best knowledge, this is the first known edge-coloring for which there are graphs with distinct values of the chromatic index and its list version.</div><div>In relation to the above, we also initiate the study of the list version of the normal edge-coloring. A <em>normal edge-coloring</em> of a cubic graph is a proper edge-coloring, in which every edge is adjacent to edges colored with 4 distinct colors or to edges colored with 2 distinct colors. It is conjectured that 5 colors suffice for a normal edge-coloring of any bridgeless cubic graph and this statement is equivalent to the Petersen Coloring Conjecture.</div><div>It turns out that similarly to strong edge-coloring, list normal edge-coloring is much more restrictive and consequently for many graphs the list normal chromatic index is greater than the normal chromatic index. In particular, we show that there are cubic graphs with list normal chromatic index at least 9, there are bridgeless cubic graphs with its value at least 8, and there are cyclically 4-edge-connected cubic graphs with value at least 7.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104243"},"PeriodicalIF":0.9,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049186","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 asymptotic uniform distribution of subset sums","authors":"Jing Wang","doi":"10.1016/j.ejc.2025.104239","DOIUrl":"10.1016/j.ejc.2025.104239","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a finite abelian group of order <span><math><mi>n</mi></math></span>, and for each <span><math><mrow><mi>a</mi><mo>∈</mo><mi>G</mi></mrow></math></span> and integer <span><math><mrow><mn>1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><mi>n</mi></mrow></math></span> let <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>a</mi></mrow></msub><mrow><mo>(</mo><mi>h</mi><mo>)</mo></mrow></mrow></math></span> denote the family of all <span><math><mi>h</mi></math></span>-element subsets of <span><math><mi>G</mi></math></span> whose sum is <span><math><mi>a</mi></math></span>. A problem posed by Katona and Makar-Limanov is to determine whether the minimum and maximum sizes of the families <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>a</mi></mrow></msub><mrow><mo>(</mo><mi>h</mi><mo>)</mo></mrow></mrow></math></span> (as <span><math><mi>a</mi></math></span> ranges over <span><math><mi>G</mi></math></span>) become asymptotically equal as <span><math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span> when <span><math><mrow><mi>h</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></math></span>. We affirmatively answer this question and in fact show that the same asymptotic equality holds for every <span><math><mrow><mn>4</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104239"},"PeriodicalIF":0.9,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145020225","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":"Palindromic length of infinite aperiodic words","authors":"Josef Rukavicka","doi":"10.1016/j.ejc.2025.104237","DOIUrl":"10.1016/j.ejc.2025.104237","url":null,"abstract":"<div><div>The palindromic length of the finite word <span><math><mi>v</mi></math></span> is equal to the minimal number of palindromes whose concatenation is equal to <span><math><mi>v</mi></math></span>. It was conjectured in 2013 that for every infinite aperiodic word <span><math><mi>x</mi></math></span>, the palindromic length of its factors is not bounded. We prove this conjecture to be true.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104237"},"PeriodicalIF":0.9,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145009544","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":"Ramsey-type problems for tilings in dense graphs","authors":"József Balogh , Andrea Freschi , Andrew Treglown","doi":"10.1016/j.ejc.2025.104228","DOIUrl":"10.1016/j.ejc.2025.104228","url":null,"abstract":"<div><div>Given a graph <span><math><mi>H</mi></math></span>, the Ramsey number <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is the smallest positive integer <span><math><mi>n</mi></math></span> such that every 2-edge-colouring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> yields a monochromatic copy of <span><math><mi>H</mi></math></span>. We write <span><math><mrow><mi>m</mi><mi>H</mi></mrow></math></span> to denote the union of <span><math><mi>m</mi></math></span> vertex-disjoint copies of <span><math><mi>H</mi></math></span>. The members of the family <span><math><mrow><mo>{</mo><mi>m</mi><mi>H</mi><mo>:</mo><mi>m</mi><mo>≥</mo><mn>1</mn><mo>}</mo></mrow></math></span> are also known as <span><math><mi>H</mi></math></span>-tilings. A well-known result of Burr, Erdős and Spencer states that <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>m</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>5</mn><mi>m</mi></mrow></math></span> for every <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span>. On the other hand, Moon proved that every 2-edge-colouring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mi>m</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span> yields a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-tiling consisting of <span><math><mi>m</mi></math></span> monochromatic copies of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, for every <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span>. Crucially, in Moon’s result, distinct copies of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> might receive different colours.</div><div>In this paper, we investigate the analogous questions where the complete host graph is replaced by a graph of large minimum degree. We determine the (asymptotic) minimum degree threshold for forcing a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-tiling covering a prescribed proportion of the vertices in a <span><math><mn>2</mn></math></span>-edge-coloured graph such that every copy of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> in the tiling is monochromatic. We also determine the largest size of a monochromatic <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-tiling one can guarantee in any 2-edge-coloured graph of large minimum degree. These results therefore provide generalisations of the theorems of Moon and Burr–Erdős–Spencer to the setting of dense graphs.</div><div>It is also natural to consider generalisations of these problems to <span><math><mi>r</mi></math></span>-edge-colourings (for <span><math><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></math></span>) and for <span><math><mi>H</mi></math></sp","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"131 ","pages":"Article 104228"},"PeriodicalIF":0.9,"publicationDate":"2025-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144926739","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}
Ron M. Adin , Arkady Berenstein , Jacob Greenstein , Jian-Rong Li , Avichai Marmor , Yuval Roichman
{"title":"Transitive and Gallai colorings of the complete graph","authors":"Ron M. Adin , Arkady Berenstein , Jacob Greenstein , Jian-Rong Li , Avichai Marmor , Yuval Roichman","doi":"10.1016/j.ejc.2025.104225","DOIUrl":"10.1016/j.ejc.2025.104225","url":null,"abstract":"<div><div>A Gallai coloring of the complete graph is an edge-coloring with no rainbow triangle. This concept first appeared in the study of incomparability graphs and anti-Ramsey theory. A directed analogue, called transitive coloring, was introduced by Berenstein, Greenstein and Li in a rather general setting. It is studied here for the acyclic tournament. The interplay of the two notions yields new enumerative results and algebraic perspectives.</div><div>We first count Gallai and transitive colorings of the complete graph which use the maximal number of colors. The quasisymmetric generating functions of these colorings, equipped with a natural descent set, are shown to be Schur-positive for any number of colors. Explicit Schur expansions are described when the number of colors is maximal. It follows that descent sets of maximal Gallai and transitive colorings are equidistributed with descent sets of perfect matchings and pattern-avoiding indecomposable permutations, respectively.</div><div>Corresponding commutative algebras are also studied. Their dimensions are shown to be equal to the number of Gallai colorings of the complete graph and the number of transitive colorings of the acyclic tournament, respectively. Relations to Orlik-Terao algebras are established.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104225"},"PeriodicalIF":0.9,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144878330","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":"Decomposition of triangle-free planar graphs","authors":"Rongxing Xu , Xuding Zhu","doi":"10.1016/j.ejc.2025.104227","DOIUrl":"10.1016/j.ejc.2025.104227","url":null,"abstract":"<div><div>A decomposition of a graph <span><math><mi>G</mi></math></span> is a family of subgraphs of <span><math><mi>G</mi></math></span> whose edge sets form a partition of <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In this paper, we prove that every triangle-free planar graph <span><math><mi>G</mi></math></span> can be decomposed into a 2-degenerate graph and a matching. Consequently, every triangle-free planar graph <span><math><mi>G</mi></math></span> has a matching <span><math><mi>M</mi></math></span> such that <span><math><mrow><mi>G</mi><mo>−</mo><mi>M</mi></mrow></math></span> is online 3-DP-colorable. This strengthens an earlier result in Škrekovski (1999) that every triangle-free planar graph is 1-defective 3-choosable.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104227"},"PeriodicalIF":0.9,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865280","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 exact Turán number of disjoint graphs– A generalization of Simonovits’ theorem, and beyond","authors":"Guantao Chen , Xingyu Lei , Shuchao Li","doi":"10.1016/j.ejc.2025.104226","DOIUrl":"10.1016/j.ejc.2025.104226","url":null,"abstract":"<div><div>For a given graph <span><math><mi>H</mi></math></span>, we say that a graph <span><math><mi>G</mi></math></span> is <span><math><mi>H</mi></math></span><em>-free</em> if it does not contain <span><math><mi>H</mi></math></span> as a subgraph. Let <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>) denote the maximum size (resp. spectral radius) of an <span><math><mi>n</mi></math></span>-vertex <span><math><mi>H</mi></math></span>-free graph, and <span><math><mrow><mtext>Ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><msub><mrow><mtext>Ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>) denote the set of all <span><math><mi>n</mi></math></span>-vertex <span><math><mi>H</mi></math></span>-free graphs with <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> edges (resp. spectral radius <span><math><mrow><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>). We call <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>) the <em>Turán number</em> (resp. <em>spectral Turán number</em>) of <span><math><mi>H</mi></math></span>. Suppose that we know the exact values of Turán numbers of <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span>, respectively. Can we get the exact value of the Turán number of the disjoint union of <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><mo>⋯</mo><mo>∪</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span>? Moon considered the disjoint union of complete graphs. A graph <span><math><mi>G</mi></math></span> is <em>color-critical</em> if there exists an edge <span><math><mi>e</mi></math></span> such that <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>−</mo><mi>e</mi><mo>)</mo></mrow><mo><</mo><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Simonovits extended Moon’s result to the disjoint union of <em>color-critical graphs</em> for sufficiently large <span><math><mi>n</mi></math></span>. Erdős et al. determined the Turán number of triangles sha","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104226"},"PeriodicalIF":0.9,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772968","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":"Splitter theorems for graph immersions","authors":"Matt DeVos, Mahdieh Malekian","doi":"10.1016/j.ejc.2025.104223","DOIUrl":"10.1016/j.ejc.2025.104223","url":null,"abstract":"<div><div>We establish splitter theorems for graph immersions for two families of graphs, <span><math><mi>k</mi></math></span>-edge-connected graphs, with <span><math><mi>k</mi></math></span> even, and 3-edge-connected, internally 4-edge-connected graphs. As a corollary, we prove that every 3-edge-connected, internally 4-edge-connected graph on at least seven vertices that immerses <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> also has <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span> as an immersion.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104223"},"PeriodicalIF":0.9,"publicationDate":"2025-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144763810","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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