{"title":"Tight bound on the minimum degree to guarantee graphs forbidding some odd cycles to be bipartite","authors":"Xiaoli Yuan, Yuejian Peng","doi":"10.1016/j.ejc.2025.104143","DOIUrl":"10.1016/j.ejc.2025.104143","url":null,"abstract":"<div><div>Erdős and Simonovits asked the following question: For an integer <span><math><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and a family of non-bipartite graphs <span><math><mi>H</mi></math></span>, determine the infimum of <span><math><mi>α</mi></math></span> such that any <span><math><mi>H</mi></math></span>-free <span><math><mi>n</mi></math></span>-vertex graph with minimum degree at least <span><math><mrow><mi>α</mi><mi>n</mi></mrow></math></span> has chromatic number at most <span><math><mi>r</mi></math></span>. We answer this question for <span><math><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow></math></span> and any family consisting of odd cycles. Let <span><math><mi>C</mi></math></span> be a family of odd cycles in which <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is the shortest odd cycle not in <span><math><mi>C</mi></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is the longest odd cycle in <span><math><mi>C</mi></math></span>, we show that if <span><math><mi>G</mi></math></span> is an <span><math><mi>n</mi></math></span>-vertex <span><math><mi>C</mi></math></span>-free graph with <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1000</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>8</mn></mrow></msup></mrow></math></span> and <span><math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>></mo><mo>max</mo><mrow><mo>{</mo><mi>n</mi><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mrow><mo>(</mo><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo><mn>2</mn><mi>n</mi><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>, then <span><math><mi>G</mi></math></span> is bipartite. Moreover, this bound on the minimum degree is tight.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104143"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143628533","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}
Zdeněk Dvořák, Benjamin Moore , Michaela Seifrtová, Robert Šámal
{"title":"Precoloring extension in planar near-Eulerian-triangulations","authors":"Zdeněk Dvořák, Benjamin Moore , Michaela Seifrtová, Robert Šámal","doi":"10.1016/j.ejc.2025.104138","DOIUrl":"10.1016/j.ejc.2025.104138","url":null,"abstract":"<div><div>We consider the 4-precoloring extension problem in <em>planar near-Eulerian- triangulations</em>, i.e., plane graphs where all faces except possibly for the outer one have length three, all vertices not incident with the outer face have even degree, and exactly the vertices incident with the outer face are precolored. We give a necessary topological condition for the precoloring to extend, and give a complete characterization when the outer face has length at most five and when all vertices of the outer face have odd degree and are colored using only three colors.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104138"},"PeriodicalIF":1.0,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yair Caro , Balázs Patkós , Zsolt Tuza , Máté Vizer
{"title":"Edge mappings of graphs: Turán type parameters","authors":"Yair Caro , Balázs Patkós , Zsolt Tuza , Máté Vizer","doi":"10.1016/j.ejc.2025.104140","DOIUrl":"10.1016/j.ejc.2025.104140","url":null,"abstract":"<div><div>In this paper, we address problems related to parameters concerning edge mappings of graphs. The quantity <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is defined to be the maximum number of edges in an <span><math><mi>n</mi></math></span>-vertex graph <span><math><mi>H</mi></math></span> such that there exists a mapping <span><math><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>→</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>≠</mo><mi>e</mi></mrow></math></span> for all <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> and further in all copies <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of <span><math><mi>G</mi></math></span> in <span><math><mi>H</mi></math></span> there exists <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span>. Among other results, we determine <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mi>G</mi></math></span> is a matching and <span><math><mi>n</mi></math></span> is large enough.</div><div>As a related concept, we say that <span><math><mi>H</mi></math></span> is unavoidable for <span><math><mi>G</mi></math></span> if for any mapping <span><math><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>→</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>≠</mo><mi>e</mi></mrow></math></span> there exists a copy <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of <span><math><mi>G</mi></math></span> in <span><math><mi>H</mi></math></span> such that <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>∉</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span>. The set of minimal unavoidable graphs for <span><math><mi>G</mi></math></span> is denoted by <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We prove that if <span><math><mi>F</mi></math></span> is a forest, then <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo><","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104140"},"PeriodicalIF":1.0,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Injective edge colorings of degenerate graphs and the oriented chromatic number","authors":"Peter Bradshaw , Alexander Clow , Jingwei Xu","doi":"10.1016/j.ejc.2025.104139","DOIUrl":"10.1016/j.ejc.2025.104139","url":null,"abstract":"<div><div>Given a graph <span><math><mi>G</mi></math></span>, an <em>injective edge-coloring</em> of <span><math><mi>G</mi></math></span> is a function <span><math><mrow><mi>ψ</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mi>N</mi></mrow></math></span> such that if <span><math><mrow><mi>ψ</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>=</mo><mi>ψ</mi><mrow><mo>(</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span>, then no third edge joins an endpoint of <span><math><mi>e</mi></math></span> and an endpoint of <span><math><msup><mrow><mi>e</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. The <em>injective chromatic index</em> of a graph <span><math><mi>G</mi></math></span>, written <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mo>inj</mo></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the minimum number of colors needed for an injective edge coloring of <span><math><mi>G</mi></math></span>. In this paper, we investigate the injective chromatic index of certain classes of degenerate graphs. First, we show that if <span><math><mi>G</mi></math></span> is a <span><math><mi>d</mi></math></span>-degenerate graph of maximum degree <span><math><mi>Δ</mi></math></span>, then <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mo>inj</mo></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>log</mo><mi>Δ</mi><mo>)</mo></mrow></mrow></math></span>. Next, we show that if <span><math><mi>G</mi></math></span> is a graph of Euler genus <span><math><mi>g</mi></math></span>, then <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mo>inj</mo></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>(</mo><mn>3</mn><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mi>g</mi></mrow></math></span>, which is tight when <span><math><mi>G</mi></math></span> is a clique. Finally, we show that the oriented chromatic number of a graph is at most exponential in its injective chromatic index. Using this fact, we prove that the oriented chromatic number of a graph embedded on a surface of Euler genus <span><math><mi>g</mi></math></span> has oriented chromatic number at most <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>g</mi></mrow><mrow><mn>6400</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, improving the previously known upper bound of <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>g</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>ɛ</mi></mrow></msup><mo>)</mo></mrow></mrow></msup></math></span> and resolving a conjecture of Aravind and Subramanian.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104139"},"PeriodicalIF":1.0,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalized Turán problem for a path and a clique","authors":"Xiaona Fang , Xiutao Zhu , Yaojun Chen","doi":"10.1016/j.ejc.2025.104137","DOIUrl":"10.1016/j.ejc.2025.104137","url":null,"abstract":"<div><div>Let <span><math><mi>H</mi></math></span> be a family of graphs. The generalized Turán number <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is the maximum number of copies of the clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> in any <span><math><mi>n</mi></math></span>-vertex <span><math><mi>H</mi></math></span>-free graph. In this paper, we determine the value of <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> for sufficiently large <span><math><mi>n</mi></math></span> with an exceptional case, and characterize all corresponding extremal graphs. This generalizes and strengthens the results of Katona (2024) on <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>. For the exceptional case, we obtain a tight upper bound for <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> that confirms a conjecture on <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> posed by Katona and Xiao.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104137"},"PeriodicalIF":1.0,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509433","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":"Planar Turán number of the 7-cycle","authors":"Ruilin Shi , Zach Walsh , Xingxing Yu","doi":"10.1016/j.ejc.2025.104134","DOIUrl":"10.1016/j.ejc.2025.104134","url":null,"abstract":"<div><div>The <em>planar Turán number</em> <span><math><mrow><msub><mrow><mo>ex</mo></mrow><mrow><mi>P</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> of a graph <span><math><mi>H</mi></math></span> is the maximum number of edges in an <span><math><mi>n</mi></math></span>-vertex planar graph without <span><math><mi>H</mi></math></span> as a subgraph. Let <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denote the cycle of length <span><math><mi>ℓ</mi></math></span>. The planar Turán number <span><math><mrow><msub><mrow><mo>ex</mo></mrow><mrow><mi>P</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> is known when <span><math><mrow><mi>ℓ</mi><mo>∈</mo><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>}</mo></mrow></mrow></math></span>, and is expected to behave differently when <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>11</mn></mrow></math></span>. We prove that <span><math><mrow><msub><mrow><mo>ex</mo></mrow><mrow><mi>P</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>18</mn><mi>n</mi></mrow><mrow><mn>7</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>48</mn></mrow><mrow><mn>7</mn></mrow></mfrac></mrow></math></span> for all <span><math><mrow><mi>n</mi><mo>≥</mo><mn>39</mn></mrow></math></span>, and show that equality holds for infinitely many integers <span><math><mi>n</mi></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104134"},"PeriodicalIF":1.0,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143429543","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 Turán problem of non-bipartite graphs: Forbidden books","authors":"Ruifang Liu, Lu Miao","doi":"10.1016/j.ejc.2025.104136","DOIUrl":"10.1016/j.ejc.2025.104136","url":null,"abstract":"<div><div>A book graph <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is a set of <span><math><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></math></span> triangles with a common edge, where <span><math><mrow><mi>r</mi><mo>≥</mo><mn>0</mn></mrow></math></span> is an integer. Zhai and Lin (2023) proved that for <span><math><mrow><mi>n</mi><mo>≥</mo><mfrac><mrow><mn>13</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>r</mi></mrow></math></span>, if <span><math><mi>G</mi></math></span> is a <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free graph of order <span><math><mi>n</mi></math></span>, then <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>ρ</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>, with equality if and only if <span><math><mrow><mi>G</mi><mo>≅</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>2</mn></mrow></msub></mrow></math></span>. Note that the extremal graph <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>2</mn></mrow></msub></math></span> is bipartite. Motivated by the above elegant result, we investigate the spectral Turán problem of non-bipartite <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free graphs of order <span><math><mi>n</mi></math></span>. For <span><math><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow></math></span>, Lin et al. (2021) provided a complete solution and proved a nice result: If <span><math><mi>G</mi></math></span> is a non-bipartite triangle-free graph of order <span><math><mi>n</mi></math></span>, then <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>ρ</mi><mrow><mo>(</mo><mrow><mi>S</mi><msub><mrow><mi>K</mi></mrow><mrow><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow><mo>,</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow></mrow></msub></mrow><mo>)</mo></mrow></mrow></math></span>, with equality if and only if <span><math><mrow><mi>G</mi><mo>≅</mo><mi>S</mi><msub><mrow><mi>K</mi></mrow><mrow><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow><mo>,</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow></mrow></msub></mrow></math></span>, where <span><math><mrow><mi>S</mi><msub><mrow><mi>K</mi></mrow><mrow><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow><mo>,</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow></mrow></msub></mrow></math></span> is the graph","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104136"},"PeriodicalIF":1.0,"publicationDate":"2025-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143422529","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 degree powers and counting stars in F-free graphs","authors":"Dániel Gerbner","doi":"10.1016/j.ejc.2025.104135","DOIUrl":"10.1016/j.ejc.2025.104135","url":null,"abstract":"<div><div>Given a positive integer <span><math><mi>r</mi></math></span> and a graph <span><math><mi>G</mi></math></span> with degree sequence <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, we define <span><math><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>r</mi></mrow></msubsup></mrow></math></span>. We let <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> be the largest value of <span><math><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> if <span><math><mi>G</mi></math></span> is an <span><math><mi>n</mi></math></span>-vertex <span><math><mi>F</mi></math></span>-free graph. We show that if <span><math><mi>F</mi></math></span> has a color-critical edge, then <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for a complete <span><math><mrow><mo>(</mo><mi>χ</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-partite graph <span><math><mi>G</mi></math></span> (this was known for cliques and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>). We obtain exact results for several other non-bipartite graphs and also determine <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>r</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. We also give simple proofs of multiple known results.</div><div>Our key observation is the connection to <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>, which is the largest number of copies of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> in <span><math><mi>n</mi></math></span>-vertex <span><math><mi>F</mi></math></span>-free graphs, where <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> is the star with <span><math><mi>r</mi></math></span> leaves. We explore this connection and apply methods from the study of <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><m","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104135"},"PeriodicalIF":1.0,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Non-trivial r-wise agreeing families","authors":"Peter Frankl , Andrey Kupavskii","doi":"10.1016/j.ejc.2025.104129","DOIUrl":"10.1016/j.ejc.2025.104129","url":null,"abstract":"<div><div>A family of subsets of <span><math><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></math></span> is <span><math><mi>r</mi></math></span>-wise agreeing if for any <span><math><mi>r</mi></math></span> sets from the family there is an element <span><math><mi>x</mi></math></span> that is either contained in all or contained in none of the <span><math><mi>r</mi></math></span> sets. The study of such families is motivated by questions in discrete optimization. In this paper, we determine the size of the largest non-trivial <span><math><mi>r</mi></math></span>-wise agreeing family. This can be seen as a generalization of the classical Brace–Daykin theorem.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104129"},"PeriodicalIF":1.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378443","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 Frank number and nowhere-zero flows on graphs","authors":"Jan Goedgebeur , Edita Máčajová , Jarne Renders","doi":"10.1016/j.ejc.2025.104127","DOIUrl":"10.1016/j.ejc.2025.104127","url":null,"abstract":"<div><div>An edge <span><math><mi>e</mi></math></span> of a graph <span><math><mi>G</mi></math></span> is called <em>deletable</em> for some orientation <span><math><mi>o</mi></math></span> if the restriction of <span><math><mi>o</mi></math></span> to <span><math><mrow><mi>G</mi><mo>−</mo><mi>e</mi></mrow></math></span> is a strong orientation. Inspired by a problem of Frank, in 2021 Hörsch and Szigeti proposed a new parameter for 3-edge-connected graphs, called the Frank number, which refines <span><math><mi>k</mi></math></span>-edge-connectivity. The <em>Frank number</em> is defined as the minimum number of orientations of <span><math><mi>G</mi></math></span> for which every edge of <span><math><mi>G</mi></math></span> is deletable in at least one of them. They showed that every 3-edge-connected graph has Frank number at most 7 and that in case these graphs are also 5-edge-colourable the parameter is at most 3. Here we strengthen both results by showing that every 3-edge-connected graph has Frank number at most 4 and that every graph which is 3-edge-connected and 3-edge-colourable has Frank number 2. The latter also confirms a conjecture by Barát and Blázsik. Furthermore, we prove two sufficient conditions for cubic graphs to have Frank number 2 and use them in an algorithm to computationally show that the Petersen graph is the only cyclically 4-edge-connected cubic graph up to 36 vertices having Frank number greater than 2.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104127"},"PeriodicalIF":1.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387118","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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