Discrete Mathematics最新文献

{"title":"An overpartition companion of Andrews and Keith's 2-colored q-series identity","authors":"Hunter Waldron","doi":"10.1016/j.disc.2025.114651","DOIUrl":"10.1016/j.disc.2025.114651","url":null,"abstract":"<div><div>Andrews and Keith recently produced a general Schmidt type partition theorem using a novel interpretation of Stockhofe's bijection, which they used to find new <em>q</em>-series identities. This includes an identity for a trivariate 2-colored partition generating function. In this paper, their Schmidt type theorem is further generalized akin to how Franklin classically extended Glaisher's theorem. As a consequence, we obtain a companion to Andrews and Keith's 2-colored identity for overpartitions. These identities appear to be special cases of a much more general result.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114651"},"PeriodicalIF":0.7,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144492009","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 forb-flex method for odd coloring and proper conflict-free coloring of planar graphs","authors":"James Anderson ,&nbsp;Herman Chau ,&nbsp;Eun-Kyung Cho ,&nbsp;Nicholas Crawford ,&nbsp;Stephen G. Hartke ,&nbsp;Emily Heath ,&nbsp;Owen Henderschedt ,&nbsp;Hyemin Kwon ,&nbsp;Zhiyuan Zhang","doi":"10.1016/j.disc.2025.114648","DOIUrl":"10.1016/j.disc.2025.114648","url":null,"abstract":"<div><div>We introduce a new technique useful for greedy coloring, which we call the forb-flex method, and apply it to odd coloring and proper conflict-free coloring of planar graphs. The odd chromatic number, denoted <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the smallest number of colors needed to properly color <em>G</em> such that every non-isolated vertex of <em>G</em> has a color appearing an odd number of times in its neighborhood. The proper conflict-free chromatic number, denoted <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>PCF</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the smallest number of colors needed to properly color <em>G</em> such that every non-isolated vertex of <em>G</em> has a color appearing uniquely in its neighborhood. Our new technique works by carefully counting the structures in the neighborhood of a vertex and determining if a neighbor of a vertex can be recolored at the end of a greedy coloring process to avoid conflicts. Combining this with the discharging method allows us to prove <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>PCF</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>4</mn></math></span> for planar graphs of girth at least 11, and <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>4</mn></math></span> for planar graphs of girth at least 10. These results improve upon the recent works of Cho, Choi, Kwon, and Park.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114648"},"PeriodicalIF":0.7,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144492011","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":"A note on the maximum size of the ground set of skew Bollobás systems","authors":"Yu Fang ,&nbsp;Xiaomiao Wang ,&nbsp;Tao Feng","doi":"10.1016/j.disc.2025.114650","DOIUrl":"10.1016/j.disc.2025.114650","url":null,"abstract":"&lt;div&gt;&lt;div&gt;A skew Bollobás system &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;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&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;mi&gt;i&lt;/mi&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; is a collection of &lt;em&gt;d&lt;/em&gt; pairwise disjoint subsets of &lt;span&gt;&lt;math&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; such that for any &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, there exist &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mo&gt;∅&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Denote by &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mtext&gt;s&lt;/mtext&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&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;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&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;/math&gt;&lt;/span&gt; the maximum size of the ground set &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;⋃&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;⋃&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; of a skew Bollobás system &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mi&gt;i&lt;/mi&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; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. We show that for any positive integers &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&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;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;,&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mtext&gt;s&lt;/mtext&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&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;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&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;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;munder&gt;&lt;mo&gt;∑","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114650"},"PeriodicalIF":0.7,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321457","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":"An optimal construction for complete graph embeddings with duals of low connectivity","authors":"Timothy Sun","doi":"10.1016/j.disc.2025.114628","DOIUrl":"10.1016/j.disc.2025.114628","url":null,"abstract":"<div><div>We describe a construction for embeddings of complete graphs where the dual has a cutvertex and the genus of the embedding is close to the minimum genus of the primal graph. When the number of vertices is congruent to 2 or 5 modulo 12, we further guarantee that the dual is simple and that the genus of the resulting embeddings matches a lower bound of Brinkmann, Noguchi, and Van den Camp, showing that their lower bound is tight infinitely often.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114628"},"PeriodicalIF":0.7,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321455","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":"Bisections of directed graphs without complete bipartite subgraphs","authors":"Wanjuan Ma,&nbsp;Shufei Wu","doi":"10.1016/j.disc.2025.114649","DOIUrl":"10.1016/j.disc.2025.114649","url":null,"abstract":"<div><div>It is well-known that every digraph (directed graph) <em>D</em> has a directed cut of size at least <span><math><mi>e</mi><mo>(</mo><mi>D</mi><mo>)</mo><mo>/</mo><mn>4</mn></math></span>, and the constant 1/4 cannot be replaced by any larger one. In this paper, motivated by a problem of Scott (2005) <span><span>[26]</span></span> and a conjecture of Lee, Loh and Sudakov (2016) <span><span>[18]</span></span>, we study bisections of digraphs, concentrating on the situation where a large number of arcs cross the bisection in each direction. For any integers <span><math><mi>d</mi><mo>≥</mo><mi>s</mi><mo>≥</mo><mn>2</mn></math></span>, let <span><math><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>s</mi></mrow></msub></mrow><mrow><mo>→</mo></mrow></mover></math></span> denote the digraph obtained by orienting each edge of the bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span> from the part of size <em>d</em> to the other part. Let <em>D</em> be a digraph with <em>m</em> arcs and minimum outdegree at least <em>d</em>. We prove that if <em>D</em> does not contain <span><math><mover><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>s</mi></mrow></msub></mrow><mrow><mo>→</mo></mrow></mover></math></span>, then <em>D</em> admits a bisection in which at least <span><math><mrow><mo>(</mo><mfrac><mrow><mn>2</mn><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>2</mn><mi>d</mi><mi>s</mi><mo>+</mo><mn>2</mn><mi>d</mi><mo>−</mo><mi>s</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>4</mn><mi>d</mi><mo>(</mo><mn>2</mn><mi>d</mi><mo>−</mo><mn>2</mn><mi>s</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow><mi>m</mi></math></span> arcs cross the bisection in each direction. Moreover, if the underlying graph of <em>D</em> does not contain triangles, we show that <em>D</em> admits a bisection in which at least <span><math><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mn>4</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow><mi>m</mi></math></span> arcs cross the bisection in each direction.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114649"},"PeriodicalIF":0.7,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321456","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":"Out of the parking lot and into the forest: Parking functions, bond lattices, and unimodal forests","authors":"Josephine Brooks ,&nbsp;Susanna Fishel ,&nbsp;Max Hlavacek ,&nbsp;Sophie Rubenfeld ,&nbsp;Bianca Carmelita Teves","doi":"10.1016/j.disc.2025.114646","DOIUrl":"10.1016/j.disc.2025.114646","url":null,"abstract":"<div><div>Rota introduced the bond lattice of a graph in <span><span>[11]</span></span>. It's a sublattice of the set partition lattice. For certain graphs, such as triangulation graphs, it's a sublattice of the important and oft studied noncrossing partition lattice. Parking functions are another central object in algebraic combinatorics. Stanley made the connection between them by defining a bijection from maximal chains of the noncrossing partition lattice to parking functions <span><span>[14]</span></span>. Motivated by Stanley's bijection, we study the maximal chains in the bond lattices of triangulation graphs.</div><div>The number of maximal chains in the bond lattice of a triangulation graph is the number of ordered cycle decompositions <span><span>[1]</span></span>, as well as being the number of rooted unimodal forests <span><span>[2]</span></span>. In this paper, we find a recursive bijection between these maximal chains and rooted unimodal forests, based on a simpler recursion than that given in <span><span>[1]</span></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114646"},"PeriodicalIF":0.7,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321453","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":"Two new families of linear codes with five Lee-weights over Fq+uFq and their Gray images","authors":"Yun Ding,&nbsp;Shixin Zhu","doi":"10.1016/j.disc.2025.114643","DOIUrl":"10.1016/j.disc.2025.114643","url":null,"abstract":"<div><div>Linear codes with few weights have many applications in secret sharing, strongly regular graphs and association schemes. In this paper, by taking proper defining sets, we first present two new infinite families of linear codes with five Lee-weights over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>+</mo><mi>u</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and exactly determine the complete weight enumerators of their Gray images. As an application, we also show that Gray images of the two families of linear codes are two new infinite families of minimal linear codes with <span><math><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>min</mi></mrow></msub></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>max</mi></mrow></msub></mrow></mfrac><mo>&lt;</mo><mfrac><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>q</mi></mrow></mfrac></math></span>, where <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>min</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>max</mi></mrow></msub></math></span> denote the minimum and maximum nonzero weights in the code, respectively.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114643"},"PeriodicalIF":0.7,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321454","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":"Asymmetry of 2-step transit probabilities in 2-coloured regular graphs","authors":"Ron Gray,&nbsp;J. Robert Johnson","doi":"10.1016/j.disc.2025.114645","DOIUrl":"10.1016/j.disc.2025.114645","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Suppose that the vertices of a regular graph are coloured red and blue with an equal number of each (we call this a balanced colouring). Since the graph is undirected, the number of edges from a red vertex to a blue vertex is clearly the same as the number of edges from a blue vertex to a red vertex. However, if instead of edges we count walks of length 2 which do not stay within their starting colour class, then this symmetry disappears. Our aim in this paper is to investigate how extreme this asymmetry can be.&lt;/div&gt;&lt;div&gt;Our main question is: Given a &lt;em&gt;d&lt;/em&gt;-regular graph, for which pairs &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; is there a balanced colouring for which the probability that a random walk starting from a red vertex stays within the red class for at least 2 steps is &lt;em&gt;x&lt;/em&gt;, and the corresponding probability for blue is &lt;em&gt;y&lt;/em&gt;?&lt;/div&gt;&lt;div&gt;Our most general result is that for any &lt;em&gt;d&lt;/em&gt;-regular graph, these pairs lie within the convex hull of the 2&lt;em&gt;d&lt;/em&gt; points &lt;span&gt;&lt;math&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&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;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&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;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;Our main focus is the torus for which we prove both sharper bounds and existence results via constructions. In particular, for the 2-dimensional torus we show that asymptotically the region in which these pairs of probabilities can lie is exactly the convex hull of:&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114645"},"PeriodicalIF":0.7,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313529","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":"Several classes of wide minimal binary linear codes based on general Maiorana-McFarland class","authors":"Xiaoni Du ,&nbsp;Siqi Gao ,&nbsp;Wenping Yuan ,&nbsp;Xingbin Qiao","doi":"10.1016/j.disc.2025.114642","DOIUrl":"10.1016/j.disc.2025.114642","url":null,"abstract":"<div><div>Minimal linear codes have important applications in secret sharing schemes and secure two-party computation. In this paper, we extend the construction of wide minimal binary linear codes presented by Ding et al. <span><span>[8]</span></span> (2018) to a more general case. More specifically, we first construct a class of Boolean functions belonging to the general Maiorana-McFarland class with more flexible parameters. Then we provide a framework for examining the Walsh transform of the new functions via the Krawtchouk polynomial. Finally, we obtain several classes of wide minimal binary linear codes with a few weights and determine their weight distribution explicitly. Our results cover all the related existing ones.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114642"},"PeriodicalIF":0.7,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313530","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":"Enumeration of sets of equiangular lines with common angle arccos⁡(1/3)","authors":"Kiyoto Yoshino","doi":"10.1016/j.disc.2025.114647","DOIUrl":"10.1016/j.disc.2025.114647","url":null,"abstract":"<div><div>In 2018, Szöllősi and Östergård used a computer to enumerate sets of equiangular lines with common angle <span><math><mi>arccos</mi><mo>⁡</mo><mo>(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>)</mo></math></span> in dimension 7. They observed that the numbers <span><math><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> of sets of <em>n</em> equiangular lines with common angle <span><math><mi>arccos</mi><mo>⁡</mo><mo>(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>)</mo></math></span> in dimension 7 are almost symmetric around <span><math><mi>n</mi><mo>=</mo><mn>14</mn></math></span>. In this paper, we prove without a computer that the numbers <span><math><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> are indeed almost symmetric by considering isometries from root lattices of rank at most 8 to the root lattice <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>8</mn></mrow></msub></math></span> of rank 8 and type <em>E</em>. Also, they determined the number <span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> of sets of <em>n</em> equiangular lines with common angle <span><math><mi>arccos</mi><mo>⁡</mo><mo>(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≤</mo><mn>13</mn></math></span>. We construct all the sets of equiangular lines with common angle <span><math><mi>arccos</mi><mo>⁡</mo><mo>(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>)</mo></math></span> in dimension greater than 7 from root lattices of type <em>A</em> or <em>D</em> with the aid of switching roots. As an application, we determine the number <span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for every positive integer <em>n</em>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114647"},"PeriodicalIF":0.7,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144307727","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}