{"title":"New MDS codes of non-GRS type and NMDS codes","authors":"Yujie Zhi, Shixin Zhu","doi":"10.1016/j.disc.2025.114436","DOIUrl":"10.1016/j.disc.2025.114436","url":null,"abstract":"<div><div>Maximum distance separable (MDS) and near maximum distance separable (NMDS) codes have been widely used in various fields such as communication systems, data storage, and quantum codes due to their algebraic properties and excellent error-correcting capabilities. This paper focuses on a specific class of linear codes and establishes necessary and sufficient conditions for them to be MDS or NMDS. Additionally, we employ the well-known Schur method to demonstrate that they are non-equivalent to generalized Reed-Solomon codes.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114436"},"PeriodicalIF":0.7,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419451","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":"Wilf classes for weak ascent sequences avoiding a pair or triple of length-3 patterns","authors":"David Callan , Toufik Mansour","doi":"10.1016/j.disc.2025.114438","DOIUrl":"10.1016/j.disc.2025.114438","url":null,"abstract":"<div><div>A <em>weak ascent sequence</em> is a word <span><math><mi>π</mi><mo>=</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>π</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> over the set of nonnegative integers such that <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>0</mn></math></span> and <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mn>1</mn><mo>+</mo><mrow><mtext>weak</mtext><mi>_</mi><mtext>asc</mtext></mrow><mo>(</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>π</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span> for <span><math><mi>i</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></math></span>, where <span><math><mrow><mtext>weak</mtext><mi>_</mi><mtext>asc</mtext></mrow><mo>(</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>π</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span> is the number of <em>weak ascents</em> in the word <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>π</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>, that is, the number of two-entry factors <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>π</mi></mrow><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> such that <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>π</mi></mrow><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. Here we obtain some enumerative results for weak ascent sequences avoiding a set of two or three 3-letter patterns, leading to a conjecture for the number of Wilf equivalence classes for weak ascent sequences avoiding a pair (respectively, triple) of 3-letter patterns. The main tool is the use of generating trees. Some cases are treated using bijective methods.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114438"},"PeriodicalIF":0.7,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143420446","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 multicolour version of the Erdős-Hajnal conjecture","authors":"Maria Axenovich, Lea Weber","doi":"10.1016/j.disc.2025.114437","DOIUrl":"10.1016/j.disc.2025.114437","url":null,"abstract":"<div><div>Informally, the multicolour version of the Erdős-Hajnal conjecture (shortly EH-conjecture) asserts that if a sufficiently large host clique on <em>n</em> vertices is edge-coloured avoiding a copy of some fixed edge-coloured clique, then there is a large homogeneous set of size <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>β</mi></mrow></msup></math></span> for some positive <em>β</em>, where a set of vertices is homogeneous if it does not induce all the colours. This conjecture, if true, claims that imposing local conditions on edge-partitions of cliques results in a global structural consequence such as a large homogeneous set, a set avoiding all edges of some part. While this conjecture attracted a lot of attention, it is still open even for two colours.</div><div>In this note, we reduce the multicolour version of the EH-conjecture to the case when the number of colours used in a host clique is either the same as in the forbidden pattern or one more. We exhibit a non-monotonicity behaviour of homogeneous sets in coloured cliques with forbidden patterns by showing that allowing an extra colour in the host graph could actually decrease the size of a largest homogeneous set.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114437"},"PeriodicalIF":0.7,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395269","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":"Enumerating alternating-runs with sign in Type A and Type B Coxeter groups","authors":"Hiranya Kishore Dey , Sivaramakrishnan Sivasubramanian","doi":"10.1016/j.disc.2025.114439","DOIUrl":"10.1016/j.disc.2025.114439","url":null,"abstract":"<div><div>We enumerate alternating runs in the alternating group <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊆</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. This leads us to enumerate the bivariate peak and valley polynomial with sign taken into account. We prove an exact formula for this signed enumerator in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and show that this polynomial depends on the value of <span><math><mi>n</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>. If <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math></span> is the polynomial enumerating alternating runs in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, Wilf showed that <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></math></span> divides <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math></span> and determined the exponent of <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></math></span> that divides <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math></span>. Let <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></math></span> be the polynomials enumerating alternating runs in <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> respectively. By finding the exponent of <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></math></span> that divides <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>±</mo></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></math></span>, we refine Wilfs result when <span><math><mi>n</mi><mo>≡</mo><mn>2</mn><mo>,</mo><mn>3</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>. When <span><math><mi>n</mi><mo>≡</mo><mn>0</mn><mo>,</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>, we show that the exponent is one short of what Wilf obtains.</div><div>As applications of our results, we get moment type identities involving the coefficients of <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>±</mo></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></math></span>, refinements to enumerating alternating and unimodal permutations in <span><math><msub><mrow><mi>A</","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114439"},"PeriodicalIF":0.7,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395268","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 the complement of a signed graph","authors":"Matteo Cavaleri, Alfredo Donno, Stefano Spessato","doi":"10.1016/j.disc.2025.114433","DOIUrl":"10.1016/j.disc.2025.114433","url":null,"abstract":"<div><div>Given a signed graph <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> and a spanning tree of Γ, we define pseudo-potentials on Γ, which coincide with usual potential functions in the balanced case. Using a pseudo-potential, we are able to define a signature on the complement <span><math><msup><mrow><mi>Γ</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span> of Γ, in such a way that the signed complete graph obtained by taking the union of Γ and <span><math><msup><mrow><mi>Γ</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span> is stable under switching equivalence, and providing a solution to an open problem in the literature of signed graphs. Then, we introduce three new notions of signed regularity, we characterize them in terms of the adjacency matrix of <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>, and we show that under such regularity hypotheses the spectrum of the signed complete graph can be described in terms of the spectra of <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> and of its signed complement. As an application of our machinery, we define a signed version of a generalization of the classical NEPS of graphs, whose signature is stable under switching equivalence. In particular, this construction allows to give a switching stable definition of the lexicographic product of signed graphs, for which the spectrum is explicitly determined in the regular case.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114433"},"PeriodicalIF":0.7,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403719","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 bipartite Turán problems on linear hypergraphs","authors":"Chuan-Ming She , Yi-Zheng Fan , Liying Kang","doi":"10.1016/j.disc.2025.114435","DOIUrl":"10.1016/j.disc.2025.114435","url":null,"abstract":"<div><div>Let <em>F</em> be a graph, and let <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> be the class of <em>r</em>-uniform Berge-<em>F</em> hypergraphs. In this paper, we establish a relationship between the spectral radius of the adjacency tensor of a uniform hypergraph and its local structure through walks. Based on the relationship, we give a spectral asymptotic bound for <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span>-free linear <em>r</em>-uniform hypergraphs and upper bounds for the spectral radii of <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo></math></span>-free or <span><math><mo>{</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo><mo>}</mo></math></span>-free linear <em>r</em>-uniform hypergraphs, where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> are respectively the triangle and the complete bipartite graph with one part having <em>s</em> vertices and the other part having <em>t</em> vertices. Our work implies an upper bound for the number of edges of <span><math><mo>{</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo><mo>}</mo></math></span>-free linear <em>r</em>-uniform hypergraphs and extends some of the existing research on (spectral) extremal problems of hypergraphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114435"},"PeriodicalIF":0.7,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395267","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}
Tianjiao Dai , Hao Li , Yannis Manoussakis , Qiancheng Ouyang
{"title":"Properly colored cycles in edge-colored complete graphs","authors":"Tianjiao Dai , Hao Li , Yannis Manoussakis , Qiancheng Ouyang","doi":"10.1016/j.disc.2025.114403","DOIUrl":"10.1016/j.disc.2025.114403","url":null,"abstract":"<div><div>As an analogy of the well-known anti-Ramsey problem, we study the existence of properly colored cycles of given length in an edge-colored complete graph. Let <span><math><mrow><mi>pr</mi></mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>G</mi><mo>)</mo></math></span> be the maximum number of colors in an edge-coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with no properly colored copy of <em>G</em>. In this paper, we determine the exact threshold for cycles <span><math><mrow><mi>pr</mi></mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span>, which proves a conjecture proposed by Fang, Győri, and Xiao, that the maximum number of colors in an edge-coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with no properly colored copy of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> is <span><math><mi>max</mi><mo></mo><mrow><mo>{</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mi>n</mi><mo>−</mo><mi>ℓ</mi><mo>+</mo><mn>1</mn><mo>,</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>⌋</mo></mrow><mi>n</mi><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mrow><mo>⌊</mo><mfrac><mrow><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>⌋</mo></mrow><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>1</mn><mo>+</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo></mrow></math></span>, where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> is a cycle on <em>ℓ</em> vertices, <span><math><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>≡</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></msub><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn></math></span>, and <span><math><mn>0</mn><mo>≤</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>≤</mo><mn>2</mn></math></span>. It is a slight modification of a previous conjecture posed by Manoussakis, Spyratos, Tuza and Voigt. Also, we consider the maximal coloring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> whether a properly colored cycle can be extended by exact one more vertex.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114403"},"PeriodicalIF":0.7,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386851","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 the domination number of graphs with minimum degree at least seven","authors":"Csilla Bujtás , Michael A. Henning","doi":"10.1016/j.disc.2025.114424","DOIUrl":"10.1016/j.disc.2025.114424","url":null,"abstract":"<div><div>A dominating set in a graph <em>G</em> is a set <span><math><mi>S</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that every vertex that is not in <em>S</em> is adjacent to a vertex from it. The domination number <span><math><mi>γ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the minimum cardinality of a dominating set in <em>G</em>. For <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>, let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> denote the class of all connected graphs with minimum degree at least <em>k</em>. The problem to determine or estimate the best possible constants <span><math><msub><mrow><mi>c</mi></mrow><mrow><mrow><mi>dom</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub></math></span> (which depend only on <em>k</em>) such that <span><math><mi>γ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mrow><mi>dom</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo>⋅</mo><mi>n</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for all <span><math><mi>G</mi><mo>∈</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is well studied, where <span><math><mi>n</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denotes the order of <em>G</em>. In this paper we establish best known lower and upper bounds to date on the constant <span><math><msub><mrow><mi>c</mi></mrow><mrow><mrow><mi>dom</mi></mrow><mo>,</mo><mn>7</mn></mrow></msub></math></span>, namely <span><math><mfrac><mrow><mn>3</mn></mrow><mrow><mn>14</mn></mrow></mfrac><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mrow><mi>dom</mi></mrow><mo>,</mo><mn>7</mn></mrow></msub><mo>≤</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>14</mn></mrow></mfrac></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114424"},"PeriodicalIF":0.7,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386852","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":"Box complexes: At the crossroad of graph theory and topology","authors":"Hamid Reza Daneshpajouh , Frédéric Meunier","doi":"10.1016/j.disc.2025.114422","DOIUrl":"10.1016/j.disc.2025.114422","url":null,"abstract":"<div><div>Various simplicial complexes can be associated with a graph. Box complexes form an important family of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their topological properties. They provide thus a fascinating topic mixing topology and discrete mathematics. This paper is intended to provide an up-do-date survey on box complexes. It is based on classical results and recent findings from the literature, but also establishes new results improving our current understanding of the topic, and identifies several challenging open questions.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114422"},"PeriodicalIF":0.7,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386853","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":"Stabilizers of consistent walks","authors":"Maruša Lekše","doi":"10.1016/j.disc.2025.114408","DOIUrl":"10.1016/j.disc.2025.114408","url":null,"abstract":"<div><div>A walk of length <em>n</em> in a graph is consistent if there exists an automorphism of the graph that maps the initial <em>n</em> vertices to the final <em>n</em> vertices of the walk, preserving their order. In this paper we find some sufficient conditions for a consistent walk in an arc-transitive graph to have trivial pointwise stabilizer. We show that in that case, the size of the smallest generating set of the automorphism group is bounded by the valence of the graph.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114408"},"PeriodicalIF":0.7,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395266","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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