Discrete Mathematics最新文献

{"title":"Upper bounds on the number of colors in interval edge-colorings of graphs","authors":"Arsen Hambardzumyan,&nbsp;Levon Muradyan","doi":"10.1016/j.disc.2024.114229","DOIUrl":"10.1016/j.disc.2024.114229","url":null,"abstract":"<div><p>An edge-coloring of a graph <em>G</em> with colors <span><math><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>t</mi></math></span> is called an <em>interval t-coloring</em> if all colors are used and the colors of edges incident to each vertex of <em>G</em> are distinct and form an interval of integers. In 1990, Kamalian proved that if a graph <em>G</em> with at least one edge has an interval <em>t</em>-coloring, then <span><math><mi>t</mi><mo>≤</mo><mn>2</mn><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>−</mo><mn>3</mn></math></span>. In 2002, Axenovich improved this upper bound for planar graphs: if a planar graph <em>G</em> admits an interval <em>t</em>-coloring, then <span><math><mi>t</mi><mo>≤</mo><mfrac><mrow><mn>11</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math></span>. In the same paper Axenovich suggested a conjecture that if a planar graph <em>G</em> has an interval <em>t</em>-coloring, then <span><math><mi>t</mi><mo>≤</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math></span>. In this paper we first prove that if a graph <em>G</em> has an interval <em>t</em>-coloring, then <span><math><mi>t</mi><mo>≤</mo><mfrac><mrow><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>+</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. Next, we confirm Axenovich's conjecture by showing that if a planar graph <em>G</em> admits an interval <em>t</em>-coloring, then <span><math><mi>t</mi><mo>≤</mo><mfrac><mrow><mn>3</mn><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>−</mo><mn>4</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. We also prove that if an outerplanar graph <em>G</em> has an interval <em>t</em>-coloring, then <span><math><mi>t</mi><mo>≤</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>−</mo><mn>1</mn></math></span>. Moreover, all these upper bounds are sharp.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114229"},"PeriodicalIF":0.7,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003601/pdfft?md5=2670a9a013dc9c49ade5c7e4ef9faf8f&pid=1-s2.0-S0012365X24003601-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142089024","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":"Subfield codes of CD-codes over F2[x]/〈x3−x〉","authors":"Anuj Kumar Bhagat,&nbsp;Ritumoni Sarma,&nbsp;Vidya Sagar","doi":"10.1016/j.disc.2024.114223","DOIUrl":"10.1016/j.disc.2024.114223","url":null,"abstract":"&lt;div&gt;&lt;p&gt;A non-zero &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-linear map from a finite-dimensional commutative &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-algebra to the field &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is called an &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-valued trace if its kernel does not contain any non-zero ideals. In this article, we utilize an &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;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-valued trace of the &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;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-algebra &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&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;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&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;mi&gt;x&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; to study binary subfield code &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for each defining set &lt;em&gt;D&lt;/em&gt; derived from a certain simplicial complex. For &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, define &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mtext&gt;Supp&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&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;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&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;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&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;, a subset of &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&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;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114223"},"PeriodicalIF":0.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003546/pdfft?md5=36a0d5563d25ed5d1b3e470afcd3ea9a&pid=1-s2.0-S0012365X24003546-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084046","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 extended 1-perfect bitrades","authors":"Evgeny A. Bespalov,&nbsp;Denis S. Krotov","doi":"10.1016/j.disc.2024.114222","DOIUrl":"10.1016/j.disc.2024.114222","url":null,"abstract":"<div><p>Extended 1-perfect codes in the Hamming scheme <span><math><mi>H</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> can be equivalently defined as codes that turn to 1-perfect codes after puncturing in any coordinate, as completely regular codes with certain intersection array, as uniformly packed codes with certain weight coefficients, as diameter perfect codes with respect to a certain anticode, as distance-4 codes with certain dual distances. We define extended 1-perfect bitrades in <span><math><mi>H</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> in five different manners, corresponding to the different definitions of extended 1-perfect codes, and prove the equivalence of these definitions of extended 1-perfect bitrades. For <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>, we prove that such bitrades exist if and only if <span><math><mi>n</mi><mo>=</mo><mi>l</mi><mi>q</mi><mo>+</mo><mn>2</mn></math></span>. For any <em>q</em>, we prove the nonexistence of extended 1-perfect bitrades if <em>n</em> is odd.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114222"},"PeriodicalIF":0.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003534/pdfft?md5=7aec3e567941af5cc4f01f8b6f836939&pid=1-s2.0-S0012365X24003534-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084036","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":"Pursuit-evasion in graphs: Zombies, lazy zombies and a survivor","authors":"Prosenjit Bose ,&nbsp;Jean-Lou De Carufel ,&nbsp;Thomas Shermer","doi":"10.1016/j.disc.2024.114220","DOIUrl":"10.1016/j.disc.2024.114220","url":null,"abstract":"&lt;div&gt;&lt;p&gt;We study &lt;em&gt;zombies and survivor&lt;/em&gt;, a variant of the game of cops and robber on graphs where the single survivor plays the role of the robber and attempts to escape from the zombies that play the role of the cops. The difference is that zombies must follow an edge of a shortest path towards the survivor on their turn. Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be the smallest number of zombies required to catch the survivor on a graph &lt;em&gt;G&lt;/em&gt; with &lt;em&gt;n&lt;/em&gt; vertices. We show that there exist outerplanar graphs and visibility graphs of simple polygons such that &lt;span&gt;&lt;math&gt;&lt;mi&gt;z&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;mi&gt;Θ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. We also show that there exist maximum-degree-3 outerplanar graphs such that &lt;span&gt;&lt;math&gt;&lt;mi&gt;z&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;mi&gt;Ω&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/p&gt;&lt;p&gt;A zombie that can remain at its current vertex on its turn is called &lt;em&gt;lazy&lt;/em&gt;. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be the smallest number of &lt;em&gt;lazy zombies&lt;/em&gt; required to catch the survivor. The ability to remain at its current vertex on its turn makes lazy zombies more powerful than normal zombies but less powerful than cops. We prove that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; for connected outerplanar graphs which is tight in the worst case. We also show that in this case, the survivor is caught after &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; rounds. We then show that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for connected graphs with treedepth &lt;em&gt;k&lt;/em&gt; and that &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; rounds are sufficient to catch the survivor. The bound on treedepth implies that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is at most &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for connected graphs with treewidth &lt;em&gt;k&lt;/em&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for connected planar graphs, &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for connected graphs with genus &lt;em&gt;g&lt;/em&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for connected graphs with any excluded &lt;em&gt;h&lt;/em&gt;-vertex minor. Our results on lazy zombies still hold when an adversary chooses ","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114220"},"PeriodicalIF":0.7,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003510/pdfft?md5=bb277c6c089d1c05a249b293a26a64fe&pid=1-s2.0-S0012365X24003510-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142077548","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":"Fractional revival on Cayley graphs over abelian groups","authors":"Jing Wang ,&nbsp;Ligong Wang ,&nbsp;Xiaogang Liu","doi":"10.1016/j.disc.2024.114218","DOIUrl":"10.1016/j.disc.2024.114218","url":null,"abstract":"<div><p>In this paper, we investigate the existence of fractional revival on Cayley graphs over finite abelian groups. We give a necessary and sufficient condition for Cayley graphs over finite abelian groups to have fractional revival. As applications, the existence of fractional revival on circulant graphs and cubelike graphs are characterized.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"347 12","pages":"Article 114218"},"PeriodicalIF":0.7,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142040369","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":"Constructing flag-transitive, point-primitive 2-designs from complete graphs","authors":"Chuyi Zhong,&nbsp;Shenglin Zhou","doi":"10.1016/j.disc.2024.114217","DOIUrl":"10.1016/j.disc.2024.114217","url":null,"abstract":"<div><p>In this paper, we study 2-designs <span><math><mi>D</mi><mo>=</mo><mo>(</mo><mi>P</mi><mo>,</mo><msup><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo>)</mo></math></span>, where <span><math><mi>P</mi></math></span> can be viewed as the edge set of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and <em>B</em> can be identified as the edge set of a subgraph of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We give a necessary condition for <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> to be flag-transitive, and then present three ways to construct such 2-designs admitting a flag-transitive, point-primitive automorphism group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. As an application, all pairs <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>G</mi><mo>)</mo></math></span> are determined, where <span><math><mi>D</mi></math></span> is a 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> design with <span><math><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>v</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>3</mn></math></span> or 4, and <em>G</em> is flag-transitive with <span><math><mi>S</mi><mi>o</mi><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>. Furthermore, we show that there are infinite flag-transitive, point-primitive 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> designs with <span><math><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>v</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>≤</mo><msup><mrow><mo>(</mo><mi>v</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span> and alternating socle <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <span><math><mi>v</mi><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114217"},"PeriodicalIF":0.7,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003480/pdfft?md5=8dd45f6c8de1e9aaa5ee26ce47fc990b&pid=1-s2.0-S0012365X24003480-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142020505","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":"Partition theorems and the Chinese Remainder Theorem","authors":"Shi-Chao Chen","doi":"10.1016/j.disc.2024.114221","DOIUrl":"10.1016/j.disc.2024.114221","url":null,"abstract":"<div><p>The famous partition theorem of Euler states that partitions of <em>n</em> into distinct parts are equinumerous with partitions of <em>n</em> into odd parts. Another famous partition theorem due to MacMahon states that the number of partitions of <em>n</em> with all parts repeated at least once equals the number of partitions of <em>n</em> where all parts must be even or congruent to <span><math><mn>3</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>6</mn><mo>)</mo></math></span>. These partition theorems were further extended by Glaisher, Andrews, Subbarao, Nyirenda and Mugwangwavari. In this paper, we utilize the Chinese Remainder Theorem to prove a comprehensive partition theorem that encompasses all existing partition theorems. We also give a natural generalization of Euler's theorem based on a special complete residue system. Furthermore, we establish interesting congruence connections between the partition function <span><math><mi>p</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and related partition functions.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"347 12","pages":"Article 114221"},"PeriodicalIF":0.7,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142013003","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":"Online size Ramsey numbers: Path vs C4","authors":"Grzegorz Adamski,&nbsp;Małgorzata Bednarska-Bzdȩga","doi":"10.1016/j.disc.2024.114214","DOIUrl":"10.1016/j.disc.2024.114214","url":null,"abstract":"<div><p>Given two graphs <em>G</em> and <em>H</em>, a size Ramsey game is played on the edge set of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span>. In every round, Builder selects an edge and Painter colours it red or blue. Builder's goal is to force Painter to create a red copy of <em>G</em> or a blue copy of <em>H</em> as soon as possible. The online (size) Ramsey number <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is the number of rounds in the game provided Builder and Painter play optimally. We prove that <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>≤</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> for every <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>. The upper bound matches the lower bound obtained by J. Cyman, T. Dzido, J. Lapinskas, and A. Lo, so we get <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>. Our proof for <span><math><mi>n</mi><mo>≤</mo><mn>13</mn></math></span> is computer-assisted. The bound <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>≤</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> solves also the “all cycles vs. <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>” game for <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span> – it implies that it takes Builder <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> rounds to force Painter to create a blue path on <em>n</em> vertices or any red cycle.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"347 12","pages":"Article 114214"},"PeriodicalIF":0.7,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142013002","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":"Further results on large sets plus of partitioned incomplete Latin squares","authors":"Hong Lu,&nbsp;Haitao Cao","doi":"10.1016/j.disc.2024.114215","DOIUrl":"10.1016/j.disc.2024.114215","url":null,"abstract":"<div><p>In this paper, we continue to study the existence of large sets plus of partitioned incomplete Latin squares of type <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mo>(</mo><mi>u</mi><mi>g</mi><mo>)</mo></mrow><mrow><mn>1</mn></mrow></msup></math></span>, denoted by LSPILS<span><math><msup><mrow></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mo>(</mo><mi>u</mi><mi>g</mi><mo>)</mo></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></math></span>. We almost solve the existence of an LSPILS<span><math><msup><mrow></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mo>(</mo><mi>u</mi><mi>g</mi><mo>)</mo></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></math></span> for any integer <span><math><mi>g</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mi>u</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn></math></span> with some possible exceptions.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114215"},"PeriodicalIF":0.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003467/pdfft?md5=a7bdcbba4d2ea5621caf6949ac6fa294&pid=1-s2.0-S0012365X24003467-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142006835","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":"Turán numbers of general star forests in hypergraphs","authors":"Lin-Peng Zhang ,&nbsp;Hajo Broersma ,&nbsp;Ligong Wang","doi":"10.1016/j.disc.2024.114219","DOIUrl":"10.1016/j.disc.2024.114219","url":null,"abstract":"<div><p>Let <span><math><mi>F</mi></math></span> be a family of <em>r</em>-uniform hypergraphs, and let <em>H</em> be an <em>r</em>-uniform hypergraph. Then <em>H</em> is called <span><math><mi>F</mi></math></span>-free if it does not contain any member of <span><math><mi>F</mi></math></span> as a subhypergraph. The Turán number of <span><math><mi>F</mi></math></span>, denoted by <span><math><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, is the maximum number of hyperedges in an <span><math><mi>F</mi></math></span>-free <em>n</em>-vertex <em>r</em>-uniform hypergraph. Our current results are motivated by earlier results on Turán numbers of star forests and hypergraph star forests. In particular, Lidický et al. (2013) <span><span>[17]</span></span> determined the Turán number <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> of a star forest <em>F</em> for sufficiently large <em>n</em>. Recently, Khormali and Palmer (2022) <span><span>[13]</span></span> generalized the above result to three different well-studied hypergraph settings (the expansions of a graph, linear hypergraphs and Berge hypergraphs), but restricted to the case that all stars in the hypergraph star forests are identical. We further generalize these results to general star forests in hypergraphs.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114219"},"PeriodicalIF":0.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003509/pdfft?md5=f55a8417dd66a400951a48477694c9f9&pid=1-s2.0-S0012365X24003509-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142006836","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}