{"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}
{"title":"Symmetries of voltage operations on polytopes, maps and maniplexes","authors":"Isabel Hubard , Elías Mochán , Antonio Montero","doi":"10.1016/j.ejc.2025.104132","DOIUrl":"10.1016/j.ejc.2025.104132","url":null,"abstract":"<div><div>Voltage operations extend traditional geometric and combinatorial operations (such as medial, truncation, prism, and pyramid over a polytope) to operations on maniplexes, maps, polytopes, and hypertopes. In classical operations, the symmetries of the original object remain in the resulting one, but sometimes additional symmetries are created; the same situation arises with voltage operations. We characterise the automorphisms of the new object that are derived from the original one and use this to bound the number of flag orbits (under the action of its automorphism group) of the new object in terms of the original one. The conditions under which the automorphism group of the original object is the same as the automorphism group of the resulting object are given. We also look at the cases where there is additional symmetry which can be accurately described due to the symmetries of the operation itself.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104132"},"PeriodicalIF":1.0,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143360670","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":"On combinatorics of string polytopes in types B and C","authors":"Yunhyung Cho , Naoki Fujita , Eunjeong Lee","doi":"10.1016/j.ejc.2025.104126","DOIUrl":"10.1016/j.ejc.2025.104126","url":null,"abstract":"<div><div>A string polytope is a rational convex polytope whose lattice points parametrize a highest weight crystal basis, which is obtained from a string cone by explicit affine inequalities depending on a highest weight. It also inherits geometric information of a flag variety such as toric degenerations, Newton–Okounkov bodies, mirror symmetry, Schubert calculus, and so on. In this paper, we study combinatorial properties of string polytopes in types <span><math><mi>B</mi></math></span> and <span><math><mi>C</mi></math></span> by giving an explicit description of string cones in these types which is analogous to Gleizer–Postnikov’s description of string cones in type <span><math><mi>A</mi></math></span>. As an application, we characterize string polytopes in type <span><math><mi>C</mi></math></span> which are unimodularly equivalent to the Gelfand–Tsetlin polytope in type <span><math><mi>C</mi></math></span> for a specific highest weight.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104126"},"PeriodicalIF":1.0,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143348629","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":"High-dimensional expanders from Kac–Moody–Steinberg groups","authors":"Laura Grave de Peralta , Inga Valentiner-Branth","doi":"10.1016/j.ejc.2025.104131","DOIUrl":"10.1016/j.ejc.2025.104131","url":null,"abstract":"<div><div>High-dimensional expanders are a generalization of the notion of expander graphs to simplicial complexes and give rise to a variety of applications in computer science and other fields. We provide a general tool to construct families of bounded degree high-dimensional spectral expanders. Inspired by the work of Kaufman and Oppenheim, we use coset complexes over quotients of Kac–Moody–Steinberg groups of rank <span><math><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></math></span>, <span><math><mi>d</mi></math></span>-spherical and purely <span><math><mi>d</mi></math></span>-spherical. We prove that infinite families of such quotients exist provided that the underlying field is of size at least 4 and the Kac–Moody–Steinberg group is 2-spherical, giving rise to new families of bounded degree high-dimensional expanders. In case the generalized Cartan matrix we consider is affine, we recover the construction of O’Donnell and Pratt from 2022 (and thus also the one by Kaufman and Oppenheim) by considering Chevalley groups as quotients of affine Kac–Moody–Steinberg groups. Moreover, our construction applies to the case where the root system is of type <span><math><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub></math></span>, a case that was not covered in earlier works.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104131"},"PeriodicalIF":1.0,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143288948","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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