{"title":"Degree conditions for disjoint path covers in digraphs","authors":"Ansong Ma, Yuefang Sun","doi":"10.1016/j.disc.2025.114410","DOIUrl":"10.1016/j.disc.2025.114410","url":null,"abstract":"<div><div>In this paper, we study degree conditions for three types of disjoint directed path cover problems: many-to-many <em>k</em>-DDPC, one-to-many <em>k</em>-DDPC and one-to-one <em>k</em>-DDPC, which are intimately connected to other famous topics in graph theory, such as Hamiltonicity and linkage.</div><div>We first get two sharp minimum semi-degree sufficient conditions for the unpaired many-to-many <em>k</em>-DDPC problem and a sharp Ore-type degree condition for the paired many-to-many 2-DDPC problem. We then obtain a minimum semi-degree sufficient condition for the one-to-many <em>k</em>-DDPC problem on a digraph with order <em>n</em>, and show that the bound for the minimum semi-degree is sharp when <span><math><mi>n</mi><mo>+</mo><mi>k</mi></math></span> is even and is sharp up to an additive constant 1 otherwise. Finally, we give a minimum semi-degree sufficient condition for the one-to-one <em>k</em>-DDPC problem on a digraph with order <em>n</em>, and show that the bound for the minimum semi-degree is sharp when <span><math><mi>n</mi><mo>+</mo><mi>k</mi></math></span> is odd and is sharp up to an additive constant 1 otherwise. Furthermore, these results hold even when <em>n</em> is (at least) a linear function of <em>k</em>. In addition, our results improve the existing results by reducing both of the lower bounds of the order and the minimum semi-degree condition of digraphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114410"},"PeriodicalIF":0.7,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143290021","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 oriented diameter of near planar triangulations","authors":"Yiwei Ge , Xiaonan Liu , Zhiyu Wang","doi":"10.1016/j.disc.2025.114406","DOIUrl":"10.1016/j.disc.2025.114406","url":null,"abstract":"<div><div>In this paper, we show that the oriented diameter of any <em>n</em>-vertex 2-connected near triangulation is at most <span><math><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span> (except for seven small exceptions), and the upper bound is tight. This extends a result of Wang et al. (2021) <span><span>[29]</span></span> on the oriented diameter of maximal outerplanar graphs, and improves an upper bound of <span><math><mi>n</mi><mo>/</mo><mn>2</mn><mo>+</mo><mi>O</mi><mo>(</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></math></span> on the oriented diameter of planar triangulations by Mondal et al. (2024) <span><span>[24]</span></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114406"},"PeriodicalIF":0.7,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143290020","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":"Three classes of propagation rules for generalized Reed-Solomon codes and their applications to EAQECCs","authors":"Ruhao Wan, Shixin Zhu","doi":"10.1016/j.disc.2025.114405","DOIUrl":"10.1016/j.disc.2025.114405","url":null,"abstract":"<div><div>In this paper, we study the Hermitian hulls of generalized Reed-Solomon (GRS) codes over finite fields. For a given class of GRS codes, by extending the length, increasing the dimension, and extending the length and increasing the dimension at the same time, we obtain three classes of GRS codes with Hermitian hulls of arbitrary dimensions. Furthermore, based on some known <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-ary Hermitian self-orthogonal GRS codes with dimension <span><math><mi>q</mi><mo>−</mo><mn>1</mn></math></span>, we obtain several classes of <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-ary maximum distance separable (MDS) codes with Hermitian hulls of arbitrary dimensions. It is worth noting that the dimension of these MDS codes can be taken from <em>q</em> to <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, and the parameters of these MDS codes can be more flexible by propagation rules. As an application, we derive three new propagation rules for MDS entanglement-assisted quantum error correction codes (EAQECCs) constructed from GRS codes. Then, from the presently known GRS codes with Hermitian hulls, we can directly obtain many MDS EAQECCs with more flexible parameters. Finally, we present several new classes of (MDS) EAQECCs with flexible parameters, and the distance of these codes can be taken from <span><math><mi>q</mi><mo>+</mo><mn>1</mn></math></span> to <span><math><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114405"},"PeriodicalIF":0.7,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143290023","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":"Eigenvalues and toughness of regular graphs","authors":"Yuanyuan Chen , Huiqiu Lin , Zhiwen Wang","doi":"10.1016/j.disc.2025.114404","DOIUrl":"10.1016/j.disc.2025.114404","url":null,"abstract":"<div><div>The toughness of a graph <em>G</em>, denoted by <span><math><mi>t</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is defined as <span><math><mi>t</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo></mo><mo>{</mo><mfrac><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mrow><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></mrow></mfrac><mo>:</mo><mi>S</mi><mo>⊂</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo><mo>></mo><mn>1</mn><mo>}</mo></math></span>. The <em>bipartite toughness</em> <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a non-complete bipartite graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> is defined as <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo></mo><mo>{</mo><mfrac><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mrow><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></mrow></mfrac><mo>:</mo><mi>S</mi><mo>⊂</mo><mi>X</mi><mspace></mspace><mtext>or</mtext><mspace></mspace><mi>S</mi><mo>⊂</mo><mi>Y</mi><mo>,</mo><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo><mo>></mo><mn>1</mn><mo>}</mo></math></span>. Incorporating the toughness and eigenvalues of a graph, we provide two sufficient eigenvalue conditions for a regular graph to be <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></mfrac><mo>−</mo></math></span>tough for a positive integer <em>b</em>, which extend a significant result by Cioabă and Wong <span><span>[10]</span></span>. For a regular bipartite graph, it is proved that <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>1</mn></math></span>. We further show a sufficient eigenvalue condition with the second largest eigenvalue for a regular bipartite graph having bipartite toughness more than 1.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114404"},"PeriodicalIF":0.7,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143290022","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}
Ling Li , Minjia Shi , Sihui Tao , Zhonghua Sun , Shixin Zhu , Jon-Lark Kim , Patrick Solé
{"title":"A generalization of the Tang-Ding binary cyclic codes","authors":"Ling Li , Minjia Shi , Sihui Tao , Zhonghua Sun , Shixin Zhu , Jon-Lark Kim , Patrick Solé","doi":"10.1016/j.disc.2024.114390","DOIUrl":"10.1016/j.disc.2024.114390","url":null,"abstract":"<div><div>Cyclic codes are an interesting family of linear codes since they have efficient decoding algorithms and contain optimal codes as subfamilies. Constructing infinite families of cyclic codes with good parameters is important in both theory and practice. Recently, Tang and Ding (2022) <span><span>[34]</span></span> proposed an infinite family of binary cyclic codes with good parameters. Shi et al. [<span><span>arXiv:2309.12003v1</span><svg><path></path></svg></span>, 2023] extended the binary Tang-Ding codes to the 4-ary case. Inspired by these two works, we study <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>s</mi></mrow></msup></math></span>-ary Tang-Ding codes, where <span><math><mi>s</mi><mo>≥</mo><mn>2</mn></math></span>. Good lower bounds on the minimum distance of the <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>s</mi></mrow></msup></math></span>-ary Tang-Ding codes are presented. As a by-product, an infinite family of <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>s</mi></mrow></msup></math></span>-ary duadic codes with a square-root like lower bound is presented.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114390"},"PeriodicalIF":0.7,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143290024","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":"Modulus for bases of matroids","authors":"Huy Truong, Pietro Poggi-Corradini","doi":"10.1016/j.disc.2025.114395","DOIUrl":"10.1016/j.disc.2025.114395","url":null,"abstract":"<div><div>In this work, we explore the application of modulus in matroid theory, specifically, the modulus of the family of bases of matroids. This study not only recovers various concepts in matroid theory, including the strength, fractional arboricity, and principal partitions, but also offers new insights. In the process, we introduce the concept of a Beurling set. Additionally, our study revisits and provides an alternative approach to two of Edmonds's theorems related to the base packing and base covering problems. This is our stepping stone for establishing Fulkerson modulus duality for the family of bases. Finally, we provide a relationship between the base modulus of matroids and their dual matroids, and a complete understanding of the base <em>p</em>-modulus across all values of <em>p</em>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114395"},"PeriodicalIF":0.7,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143290196","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":"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}
{"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}
{"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}
{"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}
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