{"title":"On the largest independent sets in the Kneser graph on chambers of PG(4,q)","authors":"Philipp Heering","doi":"10.1016/j.disc.2024.114392","DOIUrl":"10.1016/j.disc.2024.114392","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> be the graph whose vertices are the chambers of the finite projective 4-space <span><math><mi>PG</mi><mo>(</mo><mn>4</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, with two vertices being adjacent if the corresponding chambers are in general position. For <span><math><mi>q</mi><mo>≥</mo><mn>749</mn></math></span> we show that <span><math><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>2</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo><msup><mrow><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> is the independence number of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and the geometric structure of the largest independent sets is described.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114392"},"PeriodicalIF":0.7,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143289905","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":"Even pairs in Berge graphs with no balanced skew-partitions","authors":"Tara Abrishami , Maria Chudnovsky , Yaqian Tang","doi":"10.1016/j.disc.2024.114388","DOIUrl":"10.1016/j.disc.2024.114388","url":null,"abstract":"<div><div>Let <em>G</em> be a Berge graph that has no odd prism and no antihole of length at least six as an induced subgraph. We show that every such graph <em>G</em> with no balanced skew-partition is either complete or has an even pair.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114388"},"PeriodicalIF":0.7,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143221732","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":"Elementary derivations of the Rogers-Fine identity and other q-series identities","authors":"Heng Huat Chan , Song Heng Chan , Zhi-Guo Liu","doi":"10.1016/j.disc.2024.114387","DOIUrl":"10.1016/j.disc.2024.114387","url":null,"abstract":"<div><div>We begin the article with a proof of the Rogers-Fine identity. We then show that the Rogers-Fine identity implies the Rogers-Ramanujan identities as well as a new finite version of the quintuple identity. Motivated by the connections between these identities, we discover an identity which yields proofs of Rogers-Ramanujan-type identities associated with the Rogers-Ramanujan continued fraction, the Ramanujan-Göllnitz-Gordon continued fraction and Ramanujan's cubic continued fraction. We also discover a new generalization of the quintuple product identity which leads to a generalization of an identity due to R.J. Evans and a short proof of <em>q</em>-Chu-Vandermonde identity that does not require the knowledge of the <em>q</em>-binomial theorem.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114387"},"PeriodicalIF":0.7,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143352436","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}
Gregory Baimetov, Ryan Bushling, Ansel Goh, Raymond Guo, Owen Jacobs, Sean Lee
{"title":"A decomposition theorem for balanced measures","authors":"Gregory Baimetov, Ryan Bushling, Ansel Goh, Raymond Guo, Owen Jacobs, Sean Lee","doi":"10.1016/j.disc.2024.114389","DOIUrl":"10.1016/j.disc.2024.114389","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> be a connected graph. A probability measure <em>μ</em> on <em>V</em> is called <em>balanced</em> if it has the following property: if <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></math></span> denotes the “earth mover's” cost of transporting all the mass of <em>μ</em> from all over the graph to the vertex <em>v</em>, then <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>μ</mi></mrow></msub></math></span> attains its global maximum at each point in the support of <em>μ</em>. We prove a decomposition result that characterizes balanced measures as convex combinations of suitable “extremal” balanced measures that we call <em>basic</em>. An upper bound on the number of basic balanced measures on <em>G</em> follows, and an example shows that this estimate is essentially sharp.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114389"},"PeriodicalIF":0.7,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143221697","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 extremal graphs for fan graphs","authors":"Loujun Yu , Yongtao Li , Yuejian Peng","doi":"10.1016/j.disc.2024.114391","DOIUrl":"10.1016/j.disc.2024.114391","url":null,"abstract":"<div><div>A well-known result of Nosal states that a graph <em>G</em> with <em>m</em> edges and <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>></mo><msqrt><mrow><mi>m</mi></mrow></msqrt></math></span> contains a triangle. Nikiforov [Combin. Probab. Comput. 11 (2002)] extended this result to cliques by showing that if <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>></mo><msqrt><mrow><mn>2</mn><mi>m</mi><mo>(</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>r</mi><mo>)</mo></mrow></msqrt></math></span>, then <em>G</em> contains a copy of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. Let <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> be the graph obtained from a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> by adding an edge to two vertices with distance two, and let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> be the friendship graph consisting of <em>k</em> triangles that share a common vertex. Recently, Zhai, Lin and Shu [European J. Combin. 95 (2021)], Sun, Li and Wei [Discrete Math. 346 (2023)], and Li, Lu and Peng [Discrete Math. 346 (2023)] proved that if <span><math><mi>m</mi><mo>≥</mo><mn>8</mn></math></span> and <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><mn>3</mn></mrow></msqrt><mo>)</mo></math></span>, then <em>G</em> contains a copy of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> and <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, respectively, unless <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∨</mo><mfrac><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. In this paper, we give a unified extension by showing that such a graph contains a copy of <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> is the join of a vertex and a path on four vertices. Our result extends the aforementioned results since <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> and <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> a","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114391"},"PeriodicalIF":0.7,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143289906","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}
Kathie Cameron , Aristotelis Chaniotis , Celina M.H. de Figueiredo , Sophie Spirkl
{"title":"The sandwich problem for odd-hole-free and even-hole-free graphs","authors":"Kathie Cameron , Aristotelis Chaniotis , Celina M.H. de Figueiredo , Sophie Spirkl","doi":"10.1016/j.disc.2024.114383","DOIUrl":"10.1016/j.disc.2024.114383","url":null,"abstract":"<div><div>For a property <span><math><mi>P</mi></math></span> of graphs, the <span><math><mi>P</mi></math></span>-<span>Sandwich-Problem</span>, introduced by Golumbic and Shamir (1993), is the following: Given a pair of graphs <span><math><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> on the same vertex set <em>V</em>, does there exist a graph <em>G</em> such that <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>V</mi></math></span>, <span><math><mi>E</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>⊆</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>⊆</mo><mi>E</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, and <em>G</em> satisfies <span><math><mi>P</mi></math></span>? A <em>hole</em> in a graph is an induced subgraph which is a cycle of length at least four. An odd (respectively even) hole is a hole of odd (respectively even) length. Given a class of graphs <span><math><mi>C</mi></math></span> and a graph <em>G</em> we say that <em>G</em> is <span><math><mi>C</mi></math></span><em>-free</em> if it contains no induced subgraph isomorphic to a member of <span><math><mi>C</mi></math></span>. In this paper we prove that if <span><math><mi>P</mi></math></span> is the property of being odd-hole-free or the property of being even-hole-free, then the <span><math><mi>P</mi></math></span>-<span>Sandwich-Problem</span> is <span><math><mtext>NP</mtext></math></span>-complete.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114383"},"PeriodicalIF":0.7,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143221696","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":"q-Parikh matrices and q-deformed binomial coefficients of words","authors":"Antoine Renard, Michel Rigo , Markus A. Whiteland","doi":"10.1016/j.disc.2024.114381","DOIUrl":"10.1016/j.disc.2024.114381","url":null,"abstract":"<div><div>We have introduced a <em>q</em>-deformation, i.e., a polynomial in <em>q</em> with natural coefficients, of the binomial coefficient of two finite words <em>u</em> and <em>v</em> counting the number of occurrences of <em>v</em> as a subword of <em>u</em>. In this paper, we examine the <em>q</em>-deformation of Parikh matrices as introduced by Eğecioğlu in 2004.</div><div>Many classical results concerning Parikh matrices generalize to this new framework: Our first important observation is that the elements of such a matrix are in fact <em>q</em>-deformations of binomial coefficients of words. We also study their inverses and we obtain new identities about <em>q</em>-binomials.</div><div>For a finite word <em>z</em> and for the sequence <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> of prefixes of an infinite word, we show that the polynomial sequence <span><math><msub><mrow><mo>(</mo><mtable><mtr><mtd><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> converges to a formal series. We present links with additive number theory and <em>k</em>-regular sequences. In the case of a periodic word <span><math><msup><mrow><mi>u</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>, we generalize a result of Salomaa: the sequence <span><math><msub><mrow><mo>(</mo><mtable><mtr><mtd><msup><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msup></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> satisfies a linear recurrence relation with polynomial coefficients. Related to the theory of integer partition, we describe the growth and the zero set of the coefficients of the series associated with <span><math><msup><mrow><mi>u</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>.</div><div>Finally, we show that the minors of a <em>q</em>-Parikh matrix are polynomials with natural coefficients and consider a generalization of Cauchy's inequality. We also compare <em>q</em>-Parikh matrices associated with an arbitrary word with those associated with a canonical word <span><math><mn>12</mn><mo>⋯</mo><mi>k</mi></math></span> made of pairwise distinct symbols.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114381"},"PeriodicalIF":0.7,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143289904","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":"Weak star-Drazin and Drazin-star matrices","authors":"Dijana Mosić , Daochang Zhang","doi":"10.1016/j.disc.2024.114386","DOIUrl":"10.1016/j.disc.2024.114386","url":null,"abstract":"<div><div>Some systems of matrix equations weaker than existing are considered using a minimal rank weak Drazin inverse and a minimal rank right weak Drazin inverse. In order to solve new systems, we define weak star-Drazin and Drazin-star matrices which are new kinds of square matrices and generalizations of star-Drazin and Drazin-star matrices. Many characterizations and expressions of weak star-Drazin and Drazin-star matrices are proposed. As consequences, we recover known results and present new results about star-Drazin and Drazin-star matrices. Interesting special cases of weak star-Drazin and Drazin-star matrices are studied for the first time in the literature. We apply weak star-Drazin and Drazin-star matrices to solve some linear equations and present their general solution forms.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114386"},"PeriodicalIF":0.7,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171251","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}
Petr Gregor , Arturo Merino , Torsten Mütze , Francesco Verciani
{"title":"Graphs that admit a Hamilton path are cup-stackable","authors":"Petr Gregor , Arturo Merino , Torsten Mütze , Francesco Verciani","doi":"10.1016/j.disc.2024.114375","DOIUrl":"10.1016/j.disc.2024.114375","url":null,"abstract":"<div><div>Fay, Hurlbert and Tennant recently introduced a one-player game on a finite connected graph <em>G</em>, which they called cup stacking. Stacks of cups are placed at the vertices of <em>G</em>, and are transferred between vertices via stacking moves, subject to certain constraints, with the goal of stacking all cups at a single target vertex. If this is possible for every target vertex of <em>G</em>, then <em>G</em> is called <em>stackable</em>. In this paper, we prove that if <em>G</em> admits a Hamilton path, then <em>G</em> is stackable, which confirms several of the conjectures raised by Fay, Hurlbert and Tennant. Furthermore, we prove stackability for certain powers of bipartite graphs, and we construct graphs of arbitrarily large minimum degree and connectivity that do not allow stacking onto any of their vertices.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114375"},"PeriodicalIF":0.7,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143221698","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 the diameter of a super-order-commuting graph","authors":"Janko Bračič , Bojan Kuzma","doi":"10.1016/j.disc.2024.114385","DOIUrl":"10.1016/j.disc.2024.114385","url":null,"abstract":"<div><div>We answer a question about the diameter of an order-super-commuting graph on a symmetric group by studying the number-theoretical concept of <em>d</em>-complete sequences of primes in arithmetic progression.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114385"},"PeriodicalIF":0.7,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171249","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}
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