Pub Date : 2025-09-11DOI: 10.1016/j.ejc.2025.104243
Borut Lužar , Edita Máčajová , Roman Soták , Diana Švecová
A strong edge-coloring 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 strong chromatic index. 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.
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.
In relation to the above, we also initiate the study of the list version of the normal edge-coloring. A normal edge-coloring 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.
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.
{"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}
Pub Date : 2025-09-09DOI: 10.1016/j.ejc.2025.104239
Jing Wang
Let be a finite abelian group of order , and for each and integer let denote the family of all -element subsets of whose sum is . A problem posed by Katona and Makar-Limanov is to determine whether the minimum and maximum sizes of the families (as ranges over ) become asymptotically equal as when . We affirmatively answer this question and in fact show that the same asymptotic equality holds for every .
{"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}
Pub Date : 2025-09-08DOI: 10.1016/j.ejc.2025.104237
Josef Rukavicka
The palindromic length of the finite word is equal to the minimal number of palindromes whose concatenation is equal to . It was conjectured in 2013 that for every infinite aperiodic word , the palindromic length of its factors is not bounded. We prove this conjecture to be true.
{"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}
Pub Date : 2025-09-02DOI: 10.1016/j.ejc.2025.104228
József Balogh , Andrea Freschi , Andrew Treglown
<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
{"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}
Pub Date : 2025-08-21DOI: 10.1016/j.ejc.2025.104225
Ron M. Adin , Arkady Berenstein , Jacob Greenstein , Jian-Rong Li , Avichai Marmor , Yuval Roichman
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.
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.
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.
{"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}
Pub Date : 2025-08-19DOI: 10.1016/j.ejc.2025.104227
Rongxing Xu , Xuding Zhu
A decomposition of a graph is a family of subgraphs of whose edge sets form a partition of . In this paper, we prove that every triangle-free planar graph can be decomposed into a 2-degenerate graph and a matching. Consequently, every triangle-free planar graph has a matching such that is online 3-DP-colorable. This strengthens an earlier result in Škrekovski (1999) that every triangle-free planar graph is 1-defective 3-choosable.
{"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}
Pub Date : 2025-08-05DOI: 10.1016/j.ejc.2025.104226
Guantao Chen , Xingyu Lei , Shuchao Li
<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
对于给定的图H，如果图G不包含H作为子图，我们说它是无H的。设ex(n，H) (p。exsp(n，H))表示最大大小。谱半径),Ex(n，H) (resp。Exsp(n，H))表示所有边为ex（n，H）的n顶点无H图的集合。谱半径exsp(n，H))。我们称ex(n，H) (p。exp (n，H)) Turán number （p. 0）假设我们分别知道G1，…，Gk的Turán个数的确切值。我们能否得到G1∪∪Gk的Turán个数的确切值？Moon考虑了完全图的不相交并。如果存在一条边e使得χ(G−e)<χ(G)，则图G是颜色临界的。Simonovits将Moon的结果推广到足够大n的色临界图的不相交并。Erdős等人确定了Turán刚好共享一个顶点的三角形的数量。Chen等人将结果扩展到只共享一个顶点的完全图。设F是Fi的一个不相交并，其中Fi是将一个顶点与所有的顶点联结在一起得到的一个图，每个Fij都是一个色临界图。对于较大的n，我们确定了ex（n，F）和exsp（n，F）。此外，我们证明了对于这些图中的每一个F，只要n足够大，Exsp（n，F）任任（n，F），这就提供了一大类图，对Liu和Ning最近提出的一个开放问题给出了一个正答案：刻画满足Exsp（n，F）任任（n，F）的图F。
{"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}
Pub Date : 2025-08-02DOI: 10.1016/j.ejc.2025.104223
Matt DeVos, Mahdieh Malekian
We establish splitter theorems for graph immersions for two families of graphs, -edge-connected graphs, with 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 also has as an immersion.
{"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}
Pub Date : 2025-07-26DOI: 10.1016/j.ejc.2025.104224
France Gheeraert , Herman Goulet-Ouellet , Julien Leroy , Pierre Stas
Return words are a classical tool for studying shift spaces with low factor complexity. In recent years, their projection inside groups have attracted some attention, for instance in the context of dendric shift spaces, of generation of pseudorandom numbers (through the welldoc property), and of profinite invariants of shift spaces. Aiming at unifying disparate works, we introduce a notion of stability for subgroups generated by return words. Within this framework, we revisit several existing results and generalize some of them. We also study general aspects of stability, such as decidability or closure under certain operations.
{"title":"Stability properties for subgroups generated by return words","authors":"France Gheeraert , Herman Goulet-Ouellet , Julien Leroy , Pierre Stas","doi":"10.1016/j.ejc.2025.104224","DOIUrl":"10.1016/j.ejc.2025.104224","url":null,"abstract":"<div><div>Return words are a classical tool for studying shift spaces with low factor complexity. In recent years, their projection inside groups have attracted some attention, for instance in the context of dendric shift spaces, of generation of pseudorandom numbers (through the welldoc property), and of profinite invariants of shift spaces. Aiming at unifying disparate works, we introduce a notion of stability for subgroups generated by return words. Within this framework, we revisit several existing results and generalize some of them. We also study general aspects of stability, such as decidability or closure under certain operations.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104224"},"PeriodicalIF":1.0,"publicationDate":"2025-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144711391","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}
Pub Date : 2025-07-16DOI: 10.1016/j.ejc.2025.104218
Changxin Ding , Alex McDonough , Lilla Tóthmérész , Chi Ho Yuen
Every regular matroid is associated with a sandpile group, which acts simply transitively on the set of bases in various ways. Ganguly and the second author introduced the notion of consistency to describe classes of actions that respect deletion–contraction in a precise sense, and proved the consistency of rotor-routing torsors (and uniqueness thereof) for plane graphs.
In this work, we prove that the class of actions introduced by Backman, Baker, and the fourth author, is consistent for regular matroids. More precisely, we prove the consistency of its generalization given by Backman, Santos and the fourth author, and independently by the first author. This extends the above existence assertion, as well as makes progress on the goal of classifying all consistent actions.
{"title":"A consistent sandpile torsor algorithm for regular matroids","authors":"Changxin Ding , Alex McDonough , Lilla Tóthmérész , Chi Ho Yuen","doi":"10.1016/j.ejc.2025.104218","DOIUrl":"10.1016/j.ejc.2025.104218","url":null,"abstract":"<div><div>Every regular matroid is associated with a <em>sandpile group</em>, which acts simply transitively on the set of bases in various ways. Ganguly and the second author introduced the notion of <em>consistency</em> to describe classes of actions that respect deletion–contraction in a precise sense, and proved the consistency of rotor-routing torsors (and uniqueness thereof) for plane graphs.</div><div>In this work, we prove that the class of actions introduced by Backman, Baker, and the fourth author, is consistent for regular matroids. More precisely, we prove the consistency of its generalization given by Backman, Santos and the fourth author, and independently by the first author. This extends the above existence assertion, as well as makes progress on the goal of classifying all consistent actions.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104218"},"PeriodicalIF":0.9,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831559","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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