Pub Date : 2025-01-16DOI: 10.1016/j.disc.2025.114400
Sergey Kirgizov , Khaydar Nurligareev
In this work, we study the concept of self-overlapping permutations, which is related to the larger study of consecutive patterns in permutations. We show that this concept admits a simple and clear geometrical meaning, and prove that a permutation can be represented as a sequence of non-self-overlapping ones. The above structural decomposition allows us to obtain equations for the corresponding generating functions, as well as the complete asymptotic expansions for the probability that a large random permutation is (non-)self-overlapping. In particular, we show that almost all permutations are non-self-overlapping, and that the corresponding asymptotic expansion has the self-reference property: the involved coefficients count non-self-overlapping permutations once again. We also establish complete asymptotic expansions of the distributions of very tight non-self-overlapping patterns, and discuss the similarities of the non-self-overlapping permutations to other permutation building blocks, such as indecomposable and simple permutations, as well as their associated asymptotics.
{"title":"Asymptotics of self-overlapping permutations","authors":"Sergey Kirgizov , Khaydar Nurligareev","doi":"10.1016/j.disc.2025.114400","DOIUrl":"10.1016/j.disc.2025.114400","url":null,"abstract":"<div><div>In this work, we study the concept of self-overlapping permutations, which is related to the larger study of consecutive patterns in permutations. We show that this concept admits a simple and clear geometrical meaning, and prove that a permutation can be represented as a sequence of non-self-overlapping ones. The above structural decomposition allows us to obtain equations for the corresponding generating functions, as well as the complete asymptotic expansions for the probability that a large random permutation is (non-)self-overlapping. In particular, we show that almost all permutations are non-self-overlapping, and that the corresponding asymptotic expansion has the self-reference property: the involved coefficients count non-self-overlapping permutations once again. We also establish complete asymptotic expansions of the distributions of very tight non-self-overlapping patterns, and discuss the similarities of the non-self-overlapping permutations to other permutation building blocks, such as indecomposable and simple permutations, as well as their associated asymptotics.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114400"},"PeriodicalIF":0.7,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143289903","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}
Pub Date : 2025-01-15DOI: 10.1016/j.disc.2025.114393
Renzo Gómez , Juan Gutiérrez
Given a graph G, the domination number is the minimum cardinality of a dominating set in G, and the packing number is the minimum cardinality of a set of vertices whose pairwise distance is at least three. The inequality is well-known. Furthermore, Henning et al. conjectured that if G is subcubic. In this paper, we show that if G is a bipartite cubic graph. This result is obtained by showing that for this class of graphs, which improves a previous bound given by Favaron. We also show that if G is a maximal outerplanar graph, and that if G is a biconvex graph, where the latter result is tight.
{"title":"Domination and packing in graphs","authors":"Renzo Gómez , Juan Gutiérrez","doi":"10.1016/j.disc.2025.114393","DOIUrl":"10.1016/j.disc.2025.114393","url":null,"abstract":"<div><div>Given a graph <em>G</em>, 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>, and the packing number <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the minimum cardinality of a set of vertices whose pairwise distance is at least three. The inequality <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>γ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is well-known. Furthermore, Henning et al. conjectured that <span><math><mi>γ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> if <em>G</em> is subcubic. In this paper, we show that <span><math><mi>γ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>120</mn></mrow><mrow><mn>49</mn></mrow></mfrac><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> if <em>G</em> is a bipartite cubic graph. This result is obtained by showing that <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>48</mn></mrow></mfrac><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math></span> for this class of graphs, which improves a previous bound given by Favaron. We also show that <span><math><mi>γ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>3</mn><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> if <em>G</em> is a maximal outerplanar graph, and that <span><math><mi>γ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> if <em>G</em> is a biconvex graph, where the latter result is tight.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114393"},"PeriodicalIF":0.7,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143221699","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}
Pub Date : 2025-01-14DOI: 10.1016/j.disc.2025.114394
Kazumasa Nomura , Paul Terwilliger
<div><div>Let <span><math><mi>X</mi><mo>=</mo><mo>(</mo><mi>X</mi><mo>,</mo><msubsup><mrow><mo>{</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo></math></span> denote a symmetric association scheme. Fix an ordering <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> of the primitive idempotents of <span><math><mi>X</mi></math></span>, and let <em>P</em> (resp. <em>Q</em>) denote the corresponding first eigenmatrix (resp. second eigenmatrix) of <span><math><mi>X</mi></math></span>. The scheme <span><math><mi>X</mi></math></span> is said to be formally self-dual (with respect to the ordering <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>) whenever <span><math><mi>P</mi><mo>=</mo><mi>Q</mi></math></span>. We define <span><math><mi>X</mi></math></span> to be numerically self-dual (with respect to the ordering <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>) whenever the intersection numbers and Krein parameters satisfy <span><math><msubsup><mrow><mi>p</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>h</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> for <span><math><mn>0</mn><mo>≤</mo><mi>h</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>d</mi></math></span>. It is known that with respect to the ordering <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>, formal self-duality implies numerical self-duality. This raises the following question: is it possible that with respect to the ordering <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>, <span><math><mi>X</mi></math></span> is numerically self-dual but not formally self-dual? This is possible as we will show. We display an example of a symmetric association scheme and an ordering the primitive idempotents with respect to which the scheme is numerically self-dual but not formally self-dual. We have the following additional results about self-duality. Assume that <span><math><mi>X</mi></math></span> is <em>P</em>-polynomial. We show that the following are equivalent: (i) <span><math><mi>X</mi></math>
{"title":"Formal self-duality and numerical self-duality for symmetric association schemes","authors":"Kazumasa Nomura , Paul Terwilliger","doi":"10.1016/j.disc.2025.114394","DOIUrl":"10.1016/j.disc.2025.114394","url":null,"abstract":"<div><div>Let <span><math><mi>X</mi><mo>=</mo><mo>(</mo><mi>X</mi><mo>,</mo><msubsup><mrow><mo>{</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo></math></span> denote a symmetric association scheme. Fix an ordering <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> of the primitive idempotents of <span><math><mi>X</mi></math></span>, and let <em>P</em> (resp. <em>Q</em>) denote the corresponding first eigenmatrix (resp. second eigenmatrix) of <span><math><mi>X</mi></math></span>. The scheme <span><math><mi>X</mi></math></span> is said to be formally self-dual (with respect to the ordering <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>) whenever <span><math><mi>P</mi><mo>=</mo><mi>Q</mi></math></span>. We define <span><math><mi>X</mi></math></span> to be numerically self-dual (with respect to the ordering <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>) whenever the intersection numbers and Krein parameters satisfy <span><math><msubsup><mrow><mi>p</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>h</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> for <span><math><mn>0</mn><mo>≤</mo><mi>h</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>d</mi></math></span>. It is known that with respect to the ordering <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>, formal self-duality implies numerical self-duality. This raises the following question: is it possible that with respect to the ordering <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>, <span><math><mi>X</mi></math></span> is numerically self-dual but not formally self-dual? This is possible as we will show. We display an example of a symmetric association scheme and an ordering the primitive idempotents with respect to which the scheme is numerically self-dual but not formally self-dual. We have the following additional results about self-duality. Assume that <span><math><mi>X</mi></math></span> is <em>P</em>-polynomial. We show that the following are equivalent: (i) <span><math><mi>X</mi></math>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114394"},"PeriodicalIF":0.7,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143221731","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}
Pub Date : 2025-01-14DOI: 10.1016/j.disc.2025.114396
Qianqian Liu , Yaxian Zhang , Heping Zhang
In a region R consisting of unit squares, a (domino) tiling is a collection of dominoes (the union of two adjacent squares) which pave fully the region. The flip graph of R is defined on the set of all tilings of R where two tilings are adjacent if we change one from the other by a flip (a rotation of a pair of side-by-side dominoes). If R is simply-connected, then its flip graph is connected. By using homology and cohomology, Saldanha, Tomei, Casarin and Romualdo obtained a criterion to decide if two tilings are in the same component of flip graph of quadriculated surface. By a graph-theoretic method, we obtain that the flip graph of a non-bipartite quadriculated torus consists of two isomorphic components. As an application, we obtain that the forcing numbers of all perfect matchings of each non-bipartite quadriculated torus form an integer-interval. For a bipartite quadriculated torus, the components of the flip graph is more complicated, and we use homology to obtain a general lower bound for the number of components of its flip graph.
在一个由单位正方形组成的区域 R 中，（多米诺）平铺图是多米诺的集合（相邻两个正方形的结合），它们铺满了整个区域。R 的翻转图定义在 R 的所有平铺集合上，如果我们通过翻转（将一对并排的多米诺骨牌旋转 90∘）改变其中一个平铺，则两个平铺相邻。如果 R 是简单连通的，那么它的翻转图就是连通的。萨尔达尼亚、托梅、卡萨林和罗穆阿尔多利用同源性和同调性，获得了判定两个翻转图是否在四曲面翻转图的同一分量中的标准。通过图论方法，我们得到了非双面四曲面环的翻转图由两个同构分量组成。作为应用，我们得到每个非双曲二次曲面环的所有完全匹配的强制数都构成一个整数区间。对于双方四曲面环，翻转图的分量更为复杂，我们利用同源性得到了其翻转图分量数的一般下限。
{"title":"Components of domino tilings under flips in quadriculated tori","authors":"Qianqian Liu , Yaxian Zhang , Heping Zhang","doi":"10.1016/j.disc.2025.114396","DOIUrl":"10.1016/j.disc.2025.114396","url":null,"abstract":"<div><div>In a region <em>R</em> consisting of unit squares, a (domino) tiling is a collection of dominoes (the union of two adjacent squares) which pave fully the region. The flip graph of <em>R</em> is defined on the set of all tilings of <em>R</em> where two tilings are adjacent if we change one from the other by a flip (a <span><math><msup><mrow><mn>90</mn></mrow><mrow><mo>∘</mo></mrow></msup></math></span> rotation of a pair of side-by-side dominoes). If <em>R</em> is simply-connected, then its flip graph is connected. By using homology and cohomology, Saldanha, Tomei, Casarin and Romualdo obtained a criterion to decide if two tilings are in the same component of flip graph of quadriculated surface. By a graph-theoretic method, we obtain that the flip graph of a non-bipartite quadriculated torus consists of two isomorphic components. As an application, we obtain that the forcing numbers of all perfect matchings of each non-bipartite quadriculated torus form an integer-interval. For a bipartite quadriculated torus, the components of the flip graph is more complicated, and we use homology to obtain a general lower bound for the number of components of its flip graph.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114396"},"PeriodicalIF":0.7,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143352435","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}
Pub Date : 2025-01-13DOI: 10.1016/j.disc.2025.114398
Jianbing Lu , Yu Zhuang
In this paper, we show that for a non-trivial quasi-symmetric 2-design with two block intersection numbers and , if is flag-transitive and point-primitive, then G is either of affine type or almost simple type. Moreover, we prove that the socle of G cannot be an alternating group. If the socle of G is a sporadic group, then and G must be one of the following: is a 2- design with block intersection numbers and , or is a 2- design with block intersection numbers and or .
{"title":"Flag-transitive point-primitive quasi-symmetric 2-designs with block intersection numbers 0 and y ≤ 10","authors":"Jianbing Lu , Yu Zhuang","doi":"10.1016/j.disc.2025.114398","DOIUrl":"10.1016/j.disc.2025.114398","url":null,"abstract":"<div><div>In this paper, we show that for a non-trivial quasi-symmetric 2-design <span><math><mi>D</mi></math></span> with two block intersection numbers <span><math><mi>x</mi><mo>=</mo><mn>0</mn></math></span> and <span><math><mn>2</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>10</mn></math></span>, if <span><math><mi>G</mi><mo>≤</mo><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>D</mi><mo>)</mo></math></span> is flag-transitive and point-primitive, then <em>G</em> is either of affine type or almost simple type. Moreover, we prove that the socle of <em>G</em> cannot be an alternating group. If the socle of <em>G</em> is a sporadic group, then <span><math><mi>D</mi></math></span> and <em>G</em> must be one of the following: <span><math><mi>D</mi></math></span> is a 2-<span><math><mo>(</mo><mn>12</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></span> design with block intersection numbers <span><math><mn>0</mn><mo>,</mo><mn>3</mn></math></span> and <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>11</mn></mrow></msub></math></span>, or <span><math><mi>D</mi></math></span> is a 2-<span><math><mo>(</mo><mn>22</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></span> design with block intersection numbers <span><math><mn>0</mn><mo>,</mo><mn>2</mn></math></span> and <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>22</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>:</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114398"},"PeriodicalIF":0.7,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143221730","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}
Pub Date : 2025-01-13DOI: 10.1016/j.disc.2025.114397
Mourad Baïou , Francisco Barahona
We present a combinatorial algorithm for determining a maximum packing of hypertrees in a capacitated hypergraph. This is an algorithmic proof of a theorem by Frank et al. [7]. This allows the extension of several algorithms developed for graphs to hypergraphs, for the k-cut problem.
{"title":"An algorithm for packing hypertrees","authors":"Mourad Baïou , Francisco Barahona","doi":"10.1016/j.disc.2025.114397","DOIUrl":"10.1016/j.disc.2025.114397","url":null,"abstract":"<div><div>We present a combinatorial algorithm for determining a maximum packing of hypertrees in a capacitated hypergraph. This is an algorithmic proof of a theorem by Frank et al. <span><span>[7]</span></span>. This allows the extension of several algorithms developed for graphs to hypergraphs, for the <em>k</em>-cut problem.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114397"},"PeriodicalIF":0.7,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143289902","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}
Pub Date : 2025-01-13DOI: 10.1016/j.disc.2024.114392
Philipp Heering
Let be the graph whose vertices are the chambers of the finite projective 4-space , with two vertices being adjacent if the corresponding chambers are in general position. For we show that is the independence number of and the geometric structure of the largest independent sets is described.
{"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}
Pub Date : 2025-01-10DOI: 10.1016/j.disc.2024.114388
Tara Abrishami , Maria Chudnovsky , Yaqian Tang
Let G 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 G with no balanced skew-partition is either complete or has an even pair.
{"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}
Pub Date : 2025-01-09DOI: 10.1016/j.disc.2024.114387
Heng Huat Chan , Song Heng Chan , Zhi-Guo Liu
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 q-Chu-Vandermonde identity that does not require the knowledge of the q-binomial theorem.
{"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}
Pub Date : 2025-01-09DOI: 10.1016/j.disc.2024.114389
Gregory Baimetov, Ryan Bushling, Ansel Goh, Raymond Guo, Owen Jacobs, Sean Lee
Let be a connected graph. A probability measure μ on V is called balanced if it has the following property: if denotes the “earth mover's” cost of transporting all the mass of μ from all over the graph to the vertex v, then attains its global maximum at each point in the support of μ. We prove a decomposition result that characterizes balanced measures as convex combinations of suitable “extremal” balanced measures that we call basic. An upper bound on the number of basic balanced measures on G follows, and an example shows that this estimate is essentially sharp.
{"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}
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