Discrete Mathematics最新文献

{"title":"A characterization of graphs of radius-r flip-width at most 2","authors":"Yeonsu Chang ,&nbsp;Sejin Ko ,&nbsp;O-joung Kwon ,&nbsp;Myounghwan Lee","doi":"10.1016/j.disc.2024.114366","DOIUrl":"10.1016/j.disc.2024.114366","url":null,"abstract":"<div><div>The radius-<em>r</em> flip-width of a graph, for <span><math><mi>r</mi><mo>∈</mo><mi>N</mi><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span>, is a graph parameter defined in terms of a variant of the cops and robber game, called the flipper game, and it was introduced by Toruńczyk (FOCS 2023). We prove that for every <span><math><mi>r</mi><mo>∈</mo><mo>(</mo><mi>N</mi><mo>∖</mo><mo>{</mo><mn>1</mn><mo>}</mo><mo>)</mo><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span>, the class of graphs of radius-<em>r</em> flip-width at most 2 is exactly the class of (<span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>, bull, gem, co-gem)-free graphs, which are known as totally decomposable graphs with respect to bi-joins.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114366"},"PeriodicalIF":0.7,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143170180","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":"Construction of new linear codes with good parameters from group rings and skew group rings","authors":"Cong Yu,&nbsp;Shixin Zhu","doi":"10.1016/j.disc.2024.114349","DOIUrl":"10.1016/j.disc.2024.114349","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper, we use the left principal ideals of group rings and skew group rings to construct linear codes over small finite fields. We study three class of groups: Semidirect product of two cyclic groups, direct product of a cyclic group and semidirect product of two cyclic groups, wreath product of a cyclic group of order &lt;em&gt;n&lt;/em&gt; and a cyclic group of order 2. Using these groups, we can get some generator matrices over &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. Then by computer search, we obtain 18 new linear codes with parameters &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;32&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;19&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;22&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;39&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;,&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;40&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;,&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;40&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;55&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;22&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;40&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;24&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;40&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;11&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;20&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;40&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;27&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;42&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;42&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;25&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;,&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;50&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;24&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;50&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;39&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;19&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;40&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;18&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;m","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114349"},"PeriodicalIF":0.7,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171269","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":"Edge DP-coloring of planar graphs without 4-cycles and specific cycles","authors":"Patcharapan Jumnongnit ,&nbsp;Kittikorn Nakprasit ,&nbsp;Watcharintorn Ruksasakchai ,&nbsp;Pongpat Sittitrai","doi":"10.1016/j.disc.2024.114353","DOIUrl":"10.1016/j.disc.2024.114353","url":null,"abstract":"<div><div>The topic of finding sufficient conditions on graphs for their edge list chromatic number to equal their maximum degree has received significant attention in graph theory. Recently, Bernshteyn and Kostochka proposed a generalization of edge list coloring called edge DP-coloring. This development naturally leads to the investigation of similar conditions for edge DP-coloring. In this paper, we find that a graph <em>G</em> has edge DP-chromatic number equal to its maximum degree if (i) <em>G</em> is a planar graph without 4-cycles and 5-cycles, and the maximum degree of <em>G</em> is at least 6; or (ii) <em>G</em> is a planar graph without 4-, 6-cycles, and adjacent 5-cycles, and the maximum degree of <em>G</em> is at least 5. Our results generalize and strengthen previous results in the literature on edge list coloring.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114353"},"PeriodicalIF":0.7,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171266","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":"Counting polynomials with distinct zeros in Galois ring GR(p2,r)","authors":"Ying Wang ,&nbsp;Haiyan Zhou","doi":"10.1016/j.disc.2024.114354","DOIUrl":"10.1016/j.disc.2024.114354","url":null,"abstract":"<div><div>Let <span><math><mi>R</mi><mo>=</mo><mi>G</mi><mi>R</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi>r</mi><mo>)</mo></math></span> be a Galois ring of characteristic <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with cardinality <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></msup></math></span>, where <em>p</em> is a prime, and <em>ξ</em> is a root of a basic irreducible polynomial of degree <em>r</em> in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>. Let <em>k</em> be a positive integer and <em>m</em> be a integer such that <span><math><mi>k</mi><mo>+</mo><mi>m</mi><mo>⩾</mo><mn>0</mn></math></span>. Fix a polynomial <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>R</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> of degree <span><math><mi>k</mi><mo>+</mo><mi>m</mi></math></span>. For a subset <em>D</em> of <em>R</em>, let <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>s</mi><mo>)</mo><mo>=</mo><mo>♯</mo><mo>{</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>R</mi><mo>[</mo><mi>x</mi><mo>]</mo><mo>|</mo><mi>deg</mi><mo>⁡</mo><mo>(</mo><mi>g</mi><mo>)</mo><mo>⩽</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mspace></mspace><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mspace></mspace><mtext>has exactly s distinct roots in D</mtext><mo>}</mo></math></span>. In this paper, we obtain formulae for <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>s</mi><mo>)</mo></math></span> when <span><math><mi>D</mi><mo>=</mo><mo>〈</mo><mi>ξ</mi><mo>〉</mo><mo>⋃</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> and <span><math><mi>m</mi><mo>⩽</mo><mn>1</mn></math></span> and give some identities for <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>s</mi><mo>)</mo></math></span> by using the generalization of the Macilliams identity.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114354"},"PeriodicalIF":0.7,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171270","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":"Hamiltonicity of certain vertex-transitive graphs revisited","authors":"Klavdija Kutnar ,&nbsp;Dragan Marušič ,&nbsp;Andriaherimanana Sarobidy Razafimahatratra","doi":"10.1016/j.disc.2024.114350","DOIUrl":"10.1016/j.disc.2024.114350","url":null,"abstract":"<div><div>Motivated by Gregor et al. (2023) <span><span>[7]</span></span>, existence of Hamilton cycles, admitting large rotational symmetry, in certain vertex-transitive graphs is investigated. Given a graph <em>X</em> with a Hamilton cycle <em>C</em>, the <em>compression factor</em> <span><math><mi>κ</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> <em>of C</em> is the order of the largest cyclic subgroup of <span><math><mtext>Aut</mtext><mspace></mspace><mo>(</mo><mi>C</mi><mo>)</mo><mo>∩</mo><mtext>Aut</mtext><mspace></mspace><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, and the <em>Hamilton compression</em> <span><math><mi>κ</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of <em>X</em> is the maximum compression factor over all of its Hamilton cycles. It is shown that for <span><math><mi>p</mi><mo>,</mo><mi>q</mi></math></span> distinct primes, vertex-primitive graphs of order <em>pq</em> have Hamilton compression equal to <em>p</em> or <em>q</em>. In addition, for each <span><math><mi>n</mi><mo>=</mo><mn>1</mn><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>e</mi></mrow></msup></mrow></msup></math></span>, <span><math><mi>e</mi><mo>&gt;</mo><mn>1</mn></math></span>, a connected vertex-transitive graph of order 3<em>n</em> and Hamilton compression equal to <em>n</em> is constructed. As a consequence Hamilton compressions of vertex-transitive graphs of order 3<em>p</em>, <em>p</em> a prime, are determined. Similarly, Hamilton compressions of vertex-transitive graphs of order 2<em>p</em>, <em>p</em> a prime, are also computed.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114350"},"PeriodicalIF":0.7,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171264","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 extrema of graphs: Forbidden star-path forests","authors":"Yanni Zhai ,&nbsp;Xiying Yuan ,&nbsp;Lihua You","doi":"10.1016/j.disc.2024.114351","DOIUrl":"10.1016/j.disc.2024.114351","url":null,"abstract":"<div><div>A path of order <em>n</em> is denoted by <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and a star of order <em>n</em> is denoted by <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>. Recently, Fang and Yuan determined the Turán numbers of <span><math><mi>k</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi></mrow></msub></math></span> and <span><math><mi>k</mi><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>∪</mo><mn>2</mn><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> for <em>n</em> appropriately large. In this paper, we extend the results to the spectral counterpart. The graphs with maximum spectral radii among graphs containing no any copy of these three kinds of star-path forests are completely characterized.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114351"},"PeriodicalIF":0.7,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171265","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":"Monochromatic cycles in 2-edge-colored bipartite graphs with large minimum degree","authors":"Yiran Zhang ,&nbsp;Yuejian Peng","doi":"10.1016/j.disc.2024.114363","DOIUrl":"10.1016/j.disc.2024.114363","url":null,"abstract":"&lt;div&gt;&lt;div&gt;For graphs &lt;em&gt;G&lt;/em&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, we write &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;⟼&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; if any red-blue edge coloring of &lt;em&gt;G&lt;/em&gt; yields a red &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; or a blue &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. The &lt;em&gt;Ramsey number&lt;/em&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the minimum number &lt;em&gt;n&lt;/em&gt; such that the complete graph &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⟼&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. There is an interesting phenomenon that for some graphs &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; there is a number &lt;span&gt;&lt;math&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; such that for any graph &lt;em&gt;G&lt;/em&gt; of order &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with minimum degree &lt;span&gt;&lt;math&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;⟼&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. When we focus on bipartite graphs, the &lt;em&gt;bipartite Ramsey number&lt;/em&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the minimum number &lt;em&gt;n&lt;/em&gt; such that the complete bipartite graph &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⟼&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Previous known related results on cycles are on the diagonal case (&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114363"},"PeriodicalIF":0.7,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143170184","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":"Some orientation theorems for restricted DP-colorings of graphs","authors":"Ian Gossett","doi":"10.1016/j.disc.2024.114352","DOIUrl":"10.1016/j.disc.2024.114352","url":null,"abstract":"<div><div>We define <em>Z-signable</em> correspondence assignments on multigraphs, which generalize <em>good</em> correspondence assignments as introduced by Kaul and Mudrock. We introduce an auxiliary digraph that allows us to prove an Alon-Tarsi style theorem for DP-colorings from <em>Z</em>-signable correspondence assignments on multigraphs, and apply this theorem to obtain three DP-coloring analogs of the Alon-Tarsi theorem for arbitrary correspondence assignments as corollaries. We illustrate the use of these corollaries for DP-colorings on a restricted class of correspondence assignments on toroidal grids.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114352"},"PeriodicalIF":0.7,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171263","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 finite groups whose power graph is claw-free","authors":"Pallabi Manna ,&nbsp;Santanu Mandal ,&nbsp;Andrea Lucchini","doi":"10.1016/j.disc.2024.114348","DOIUrl":"10.1016/j.disc.2024.114348","url":null,"abstract":"<div><div>Let <em>G</em> be a finite group and let <span><math><mi>P</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the undirected power graph of <em>G</em>. Recall that the vertices of <span><math><mi>P</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> are labelled by the elements of <em>G</em>, with an edge between <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> if either <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><mo>〈</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>〉</mo></math></span> or <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mo>〈</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>〉</mo></math></span>. The subgraph induced by the non-identity elements is called the reduced power graph, denoted by <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. The main purpose of this paper is to investigate the finite groups whose reduced power graph is claw-free, which means that it has no vertex with three pairwise non-adjacent neighbours. In particular, we prove that if <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is claw-free, then either <em>G</em> is solvable or <em>G</em> is an almost simple group. In the second case, the socle of <em>G</em> is isomorphic to <span><math><mrow><mi>PSL</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> for suitable choices of <em>q</em>. Finally we prove that if <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is claw-free, then the order of <em>G</em> is divisible by at most 5 different primes.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114348"},"PeriodicalIF":0.7,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143170516","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":"Unavoidable induced subgraphs of infinite 2-connected graphs","authors":"Sarah Allred ,&nbsp;Guoli Ding ,&nbsp;Bogdan Oporowski","doi":"10.1016/j.disc.2024.114346","DOIUrl":"10.1016/j.disc.2024.114346","url":null,"abstract":"<div><div>In 1930, Ramsey proved that every infinite graph contains either an infinite clique or an infinite independent set. König proved that every connected infinite graph contains either a ray or a vertex of infinite degree. In this paper, we establish the 2-connected analog of these results.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114346"},"PeriodicalIF":0.7,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171267","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}