Pub Date : 2025-11-28DOI: 10.1016/j.disc.2025.114903
Ping Zhang , Honggang Hu , Peng Wang , Yiyuan Luo , Lifa Wu
How can one optimise existing permutation-based pseudorandom functions (PRFs) to minimize the number of components (such as keys, permutations, and inverse permutations) while ensuring beyond-birthday-bound (BBB) security has always been an open problem. This paper focuses on this open problem and proposes a minimal BBB secure permutation-based PRF. Based on PDMMAC, we utilize a single permutation just with forward calls instead of forward and backward calls to construct an improved single-keyed permutation-based PRF, called PDM+. To present better security analyses, we generalize the traditional sum-capture lemma to more general settings. Finally, we prove that PDM+ ensure BBB security up to about adversarial construction and primitive queries by the expectation method and generalized sum-capture lemmas.
{"title":"Minimizing beyond-birthday-bound secure permutation-based pseudorandom function","authors":"Ping Zhang , Honggang Hu , Peng Wang , Yiyuan Luo , Lifa Wu","doi":"10.1016/j.disc.2025.114903","DOIUrl":"10.1016/j.disc.2025.114903","url":null,"abstract":"<div><div>How can one optimise existing permutation-based pseudorandom functions (PRFs) to minimize the number of components (such as keys, permutations, and inverse permutations) while ensuring beyond-birthday-bound (BBB) security has always been an open problem. This paper focuses on this open problem and proposes a minimal BBB secure permutation-based PRF. Based on PDMMAC, we utilize a single permutation just with forward calls instead of forward and backward calls to construct an improved single-keyed permutation-based PRF, called PDM+. To present better security analyses, we generalize the traditional sum-capture lemma to more general settings. Finally, we prove that PDM+ ensure BBB security up to about <span><math><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>n</mi><mo>/</mo><mn>3</mn></mrow></msup></math></span> adversarial construction and primitive queries by the expectation method and generalized sum-capture lemmas.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 4","pages":"Article 114903"},"PeriodicalIF":0.7,"publicationDate":"2025-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625006","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 : 2025-11-24DOI: 10.1016/j.disc.2025.114902
Jian-Hua Yin
Given a graph H, a graphic sequence π is potentially H-graphic if there is a realization of π containing H as a subgraph. Erdős et al. introduced the following problem: determine the minimum even integer such that each n-term graphic sequence with sum at least is potentially H-graphic. The parameter is known as the potential function of H, and can be viewed as a degree sequence variant of the classical extremal function ex. Ferrara et al. (2016) [3] established an upper bound on and determined asymptotically for an arbitrary graph H. Yin (2020) [6] also obtained an upper bound on . Erbes et al. (2018) [1] defined a stability concept for the potential number, which is a natural analogue to the stability of the classical extremal function given by Simonovits. They gave a sufficient condition for a graph H to be stable with respect to the potential function, and characterized the stability of those graphs H that contain an induced subgraph of order with exactly one edge. In this paper, we further characterize the stability of those graphs H that contain an induced subgraph of order with exactly t independent edges for . Therefore, the stability for all graphs H is characterized completely.
给定一个图H，一个图序列π是潜在的H图，如果π包含H作为子图的实现。Erdős等人引入了以下问题：确定最小偶数σ(H,n)，使得每个求和至少为σ(H,n)的n项图序列都是潜在的H图。参数σ(H,n)称为H的势函数，可以看作是经典极值函数ex（n，H）的阶序列变体。Ferrara et al.(2016)[3]建立了σ(H,n)的上界，并渐近地确定了任意图H的σ(H,n)。Yin(2020)[6]也获得了σ(H,n)的上界。Erbes et al.(2018)[1]定义了势数的稳定性概念，这是Simonovits给出的经典极值函数稳定性的自然类比。他们给出了图H相对于势函数稳定的充分条件，并刻画了含有α(H)+1阶的诱导子图H只有一条边的图H的稳定性。在本文中，我们进一步刻画了含有α(H)+1阶的诱导子图H的稳定性，这些图H具有恰好t条独立边，且1≤t≤⌊α+12⌋。因此，所有图H的稳定性被完全刻画。
{"title":"A note on the stability of the potential function","authors":"Jian-Hua Yin","doi":"10.1016/j.disc.2025.114902","DOIUrl":"10.1016/j.disc.2025.114902","url":null,"abstract":"<div><div>Given a graph <em>H</em>, a graphic sequence <em>π</em> is <em>potentially H</em>-<em>graphic</em> if there is a realization of <em>π</em> containing <em>H</em> as a subgraph. Erdős et al. introduced the following problem: determine the minimum even integer <span><math><mi>σ</mi><mo>(</mo><mi>H</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> such that each <em>n</em>-term graphic sequence with sum at least <span><math><mi>σ</mi><mo>(</mo><mi>H</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> is potentially <em>H</em>-graphic. The parameter <span><math><mi>σ</mi><mo>(</mo><mi>H</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> is known as the <em>potential function</em> of <em>H</em>, and can be viewed as a degree sequence variant of the classical extremal function ex<span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>. Ferrara et al. (2016) <span><span>[3]</span></span> established an upper bound on <span><math><mi>σ</mi><mo>(</mo><mi>H</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> and determined <span><math><mi>σ</mi><mo>(</mo><mi>H</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> asymptotically for an arbitrary graph <em>H</em>. Yin (2020) <span><span>[6]</span></span> also obtained an upper bound on <span><math><mi>σ</mi><mo>(</mo><mi>H</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>. Erbes et al. (2018) <span><span>[1]</span></span> defined a stability concept for the potential number, which is a natural analogue to the stability of the classical extremal function given by Simonovits. They gave a sufficient condition for a graph <em>H</em> to be stable with respect to the potential function, and characterized the stability of those graphs <em>H</em> that contain an induced subgraph of order <span><math><mi>α</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> with exactly one edge. In this paper, we further characterize the stability of those graphs <em>H</em> that contain an induced subgraph of order <span><math><mi>α</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> with exactly <em>t</em> independent edges for <span><math><mn>1</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mo>⌊</mo><mfrac><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></math></span>. Therefore, the stability for all graphs <em>H</em> is characterized completely.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 4","pages":"Article 114902"},"PeriodicalIF":0.7,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625007","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 : 2025-11-24DOI: 10.1016/j.disc.2025.114896
Aseem Dalal, Bhawani Sankar Panda
<div><div>The total chromatic number, <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, of a graph <em>G</em> is the minimum number of colors required to totally color <em>G</em>. A graph <em>G</em> is of Type 1 is <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> and of Type 2 if <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>2</mn></math></span>. A 1992 result of Chew and Yap shows that all complete multipartite graphs of odd order are of Type 1. For graphs of even order, a long-standing conjecture by Hoffman and Rodger states that a complete multipartite graph <em>G</em> of even order is of Type 2 if and only if either it is regular bipartite, or its deficiency, <span><math><mi>d</mi><mi>e</mi><mi>f</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>)</mo></math></span>, is less than the number of parts of odd size. For a complete multipartite graph <span><math><mi>G</mi><mo>=</mo><mi>G</mi><mo>[</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>]</mo></math></span> of even order (where <span><math><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>=</mo><mo>…</mo><mo>=</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>|</mo><mo>=</mo><mi>r</mi><mo><</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>|</mo><mo>≤</mo><mo>…</mo><mo>≤</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>|</mo></math></span>), the conjecture has been verified in several cases: when <span><math><mi>m</mi><mo>≤</mo><mn>4</mn></math></span>, when <span><math><mi>p</mi><mo>≤</mo><mn>6</mn></math></span> or, when <span><math><mi>d</mi><mi>e</mi><mi>f</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is very large. In this paper, we prove that the conjecture holds for all such graphs <em>G</em> satisfying<span><span><span><math><mfrac><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></mfrac><mo><</mo><mrow><mo>{</mo><mtable><mtr><mtd><mn>1</mn><mo>/</mo><mn>2</mn></mtd><mtd><mtext>when </mtext><mi>r</mi><mtext> is even, </mtext></mtd></mtr><mtr><mtd><mn>1</mn><mo>/</mo><mn>3</mn></mtd><mtd><mtext>when </mtext><mi>r</mi><mtext> is odd, </mtext></mtd></mtr></mtable></mrow></math></span></span></span> thereb
{"title":"A sufficient condition for complete multipartite graphs to be of Type 1","authors":"Aseem Dalal, Bhawani Sankar Panda","doi":"10.1016/j.disc.2025.114896","DOIUrl":"10.1016/j.disc.2025.114896","url":null,"abstract":"<div><div>The total chromatic number, <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, of a graph <em>G</em> is the minimum number of colors required to totally color <em>G</em>. A graph <em>G</em> is of Type 1 is <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> and of Type 2 if <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>2</mn></math></span>. A 1992 result of Chew and Yap shows that all complete multipartite graphs of odd order are of Type 1. For graphs of even order, a long-standing conjecture by Hoffman and Rodger states that a complete multipartite graph <em>G</em> of even order is of Type 2 if and only if either it is regular bipartite, or its deficiency, <span><math><mi>d</mi><mi>e</mi><mi>f</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>)</mo></math></span>, is less than the number of parts of odd size. For a complete multipartite graph <span><math><mi>G</mi><mo>=</mo><mi>G</mi><mo>[</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>]</mo></math></span> of even order (where <span><math><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>=</mo><mo>…</mo><mo>=</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>|</mo><mo>=</mo><mi>r</mi><mo><</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>|</mo><mo>≤</mo><mo>…</mo><mo>≤</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>|</mo></math></span>), the conjecture has been verified in several cases: when <span><math><mi>m</mi><mo>≤</mo><mn>4</mn></math></span>, when <span><math><mi>p</mi><mo>≤</mo><mn>6</mn></math></span> or, when <span><math><mi>d</mi><mi>e</mi><mi>f</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is very large. In this paper, we prove that the conjecture holds for all such graphs <em>G</em> satisfying<span><span><span><math><mfrac><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></mfrac><mo><</mo><mrow><mo>{</mo><mtable><mtr><mtd><mn>1</mn><mo>/</mo><mn>2</mn></mtd><mtd><mtext>when </mtext><mi>r</mi><mtext> is even, </mtext></mtd></mtr><mtr><mtd><mn>1</mn><mo>/</mo><mn>3</mn></mtd><mtd><mtext>when </mtext><mi>r</mi><mtext> is odd, </mtext></mtd></mtr></mtable></mrow></math></span></span></span> thereb","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 4","pages":"Article 114896"},"PeriodicalIF":0.7,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625009","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 : 2025-11-24DOI: 10.1016/j.disc.2025.114889
Ákos G.Horváth
In this paper, we investigate the configuration theorems of Desargues and Pappus in a synthetic geometric way. We provide a bridge between the two configurations with a third one that can be considered a specification for both. We do not use the theory of collineations or the analytic description of the plane over a ternary ring.
{"title":"A bridge between Desargues' and Pappus' theorems","authors":"Ákos G.Horváth","doi":"10.1016/j.disc.2025.114889","DOIUrl":"10.1016/j.disc.2025.114889","url":null,"abstract":"<div><div>In this paper, we investigate the configuration theorems of Desargues and Pappus in a synthetic geometric way. We provide a bridge between the two configurations with a third one that can be considered a specification for both. We do not use the theory of collineations or the analytic description of the plane over a ternary ring.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 4","pages":"Article 114889"},"PeriodicalIF":0.7,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625008","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 : 2025-11-21DOI: 10.1016/j.disc.2025.114876
Eli Bagno , David Garber
Stirling numbers of the first and the second kinds have seen many generalizations and applications in various areas of mathematics. We introduce some combinatorial parameters that realize q-analogues and Broder's r-variants of Stirling numbers of type B of both kinds, which count signed set partitions and signed permutations, respectively.
{"title":"Combinatorics of q,r-analogues of Stirling numbers of type B","authors":"Eli Bagno , David Garber","doi":"10.1016/j.disc.2025.114876","DOIUrl":"10.1016/j.disc.2025.114876","url":null,"abstract":"<div><div>Stirling numbers of the first and the second kinds have seen many generalizations and applications in various areas of mathematics. We introduce some combinatorial parameters that realize <em>q</em>-analogues and Broder's <em>r</em>-variants of Stirling numbers of type <em>B</em> of both kinds, which count signed set partitions and signed permutations, respectively.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 4","pages":"Article 114876"},"PeriodicalIF":0.7,"publicationDate":"2025-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145555275","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 : 2025-11-17DOI: 10.1016/j.disc.2025.114888
Xiamiao Zhao , Yuxuan Yang
A graph G of order n is called edge-pancyclic if, for every integer k with , every edge of G lies in a cycle of length k. Determining the minimum size of a simple edge-pancyclic graph with n vertices seems difficult. Recently, Li, Liu and Zhan gave both a lower bound and an upper bound on . In this paper, we improve their lower bound by considering a new class of graphs and improve the upper bound by constructing a family of edge-pancyclic graphs.
{"title":"On the minimum size of an edge-pancyclic graph of a given order","authors":"Xiamiao Zhao , Yuxuan Yang","doi":"10.1016/j.disc.2025.114888","DOIUrl":"10.1016/j.disc.2025.114888","url":null,"abstract":"<div><div>A graph <em>G</em> of order <em>n</em> is called edge-pancyclic if, for every integer <em>k</em> with <span><math><mn>3</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>, every edge of <em>G</em> lies in a cycle of length <em>k</em>. Determining the minimum size <span><math><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> of a simple edge-pancyclic graph with <em>n</em> vertices seems difficult. Recently, Li, Liu and Zhan gave both a lower bound and an upper bound on <span><math><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. In this paper, we improve their lower bound by considering a new class of graphs and improve the upper bound by constructing a family of edge-pancyclic graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114888"},"PeriodicalIF":0.7,"publicationDate":"2025-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578778","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 : 2025-11-15DOI: 10.1016/j.disc.2025.114886
Florent Foucaud , Paras Vinubhai Maniya , Kaustav Paul , Dinabandhu Pradhan
A dominating set of a graph G is a set such that every vertex in is adjacent to at least one vertex in D. A set is a locating set of G if every two vertices in have pairwise distinct open neighborhoods in L. A set is a locating-dominating set of G if D is a dominating set and a locating set of G. The location-domination number of G, denoted by , is the minimum cardinality among all locating-dominating sets of G. A well-known conjecture in the study of locating-dominating sets is that if G is an isolate-free and twin-free graph of order n, then . Recently, Bousquet et al. (2025) [5] proved that if G is an isolate-free and twin-free graph of order n, then and posed the question whether the vertex set of such a graph can be partitioned into two locating sets. We answer this question affirmatively for twin-free distance-hereditary graphs, maximal outerplanar graphs, split graphs, and co-bipartite graphs. In fact, we prove a stronger result: for any graph G without isolated vertices and twin vertices, if G is a distance-hereditary graph or a maximal outerplanar graph or a split graph or a co-bipartite graph, then the vertex set of G can be partitioned into two locating-dominating sets. Consequently, this also confirms the original conjecture for these graph classes.
{"title":"Locating-dominating partitions for some classes of graphs","authors":"Florent Foucaud , Paras Vinubhai Maniya , Kaustav Paul , Dinabandhu Pradhan","doi":"10.1016/j.disc.2025.114886","DOIUrl":"10.1016/j.disc.2025.114886","url":null,"abstract":"<div><div>A dominating set of a graph <em>G</em> is a set <span><math><mi>D</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that every vertex in <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>∖</mo><mi>D</mi></math></span> is adjacent to at least one vertex in <em>D</em>. A set <span><math><mi>L</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is a locating set of <em>G</em> if every two vertices in <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>∖</mo><mi>L</mi></math></span> have pairwise distinct open neighborhoods in <em>L</em>. A set <span><math><mi>D</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is a locating-dominating set of <em>G</em> if <em>D</em> is a dominating set and a locating set of <em>G</em>. The location-domination number of <em>G</em>, denoted by <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>L</mi><mi>D</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the minimum cardinality among all locating-dominating sets of <em>G</em>. A well-known conjecture in the study of locating-dominating sets is that if <em>G</em> is an isolate-free and twin-free graph of order <em>n</em>, then <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>L</mi><mi>D</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. Recently, Bousquet et al. (2025) <span><span>[5]</span></span> proved that if <em>G</em> is an isolate-free and twin-free graph of order <em>n</em>, then <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>L</mi><mi>D</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>⌈</mo><mfrac><mrow><mn>5</mn><mi>n</mi></mrow><mrow><mn>8</mn></mrow></mfrac><mo>⌉</mo></math></span> and posed the question whether the vertex set of such a graph can be partitioned into two locating sets. We answer this question affirmatively for twin-free distance-hereditary graphs, maximal outerplanar graphs, split graphs, and co-bipartite graphs. In fact, we prove a stronger result: for any graph <em>G</em> without isolated vertices and twin vertices, if <em>G</em> is a distance-hereditary graph or a maximal outerplanar graph or a split graph or a co-bipartite graph, then the vertex set of <em>G</em> can be partitioned into two locating-dominating sets. Consequently, this also confirms the original conjecture for these graph classes.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114886"},"PeriodicalIF":0.7,"publicationDate":"2025-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528742","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 : 2025-11-15DOI: 10.1016/j.disc.2025.114885
Varsha Tiwari, Pramod Kumar Kewat
In this article, we utilize reversed Dickson polynomials of the first kind to construct cyclic codes and determine the exact minimum distances of these codes. As a result, we derive many families of optimal and almost-optimal cyclic codes. We also provide conditions for these codes to be dual-containing. As an application, we use these cyclic codes to construct quantum error correcting codes via the Calderbank-Shor-Steane (CSS) construction and Steane's enlargement method, resulting in new and optimal quantum error correcting codes. Additionally, we introduce quantum synchronizable codes (QSCs) with the maximum synchronization capability.
{"title":"Construction of good cyclic codes and quantum error correcting codes using reversed Dickson polynomials","authors":"Varsha Tiwari, Pramod Kumar Kewat","doi":"10.1016/j.disc.2025.114885","DOIUrl":"10.1016/j.disc.2025.114885","url":null,"abstract":"<div><div>In this article, we utilize reversed Dickson polynomials of the first kind to construct cyclic codes and determine the exact minimum distances of these codes. As a result, we derive many families of optimal and almost-optimal cyclic codes. We also provide conditions for these codes to be dual-containing. As an application, we use these cyclic codes to construct quantum error correcting codes via the Calderbank-Shor-Steane (CSS) construction and Steane's enlargement method, resulting in new and optimal quantum error correcting codes. Additionally, we introduce quantum synchronizable codes (QSCs) with the maximum synchronization capability.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114885"},"PeriodicalIF":0.7,"publicationDate":"2025-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528743","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}
For a given graph F, the r-uniform suspension of F is the r-uniform hypergraph obtained from F by taking new vertices and adding them to every edge. In this paper, we consider Turán problems on suspension hypergraphs, and we obtain several general and exact results.
{"title":"On Turán problems for suspension hypergraphs","authors":"Xin Cheng , Dániel Gerbner , Hilal Hama Karim , Junpeng Zhou","doi":"10.1016/j.disc.2025.114887","DOIUrl":"10.1016/j.disc.2025.114887","url":null,"abstract":"<div><div>For a given graph <em>F</em>, the <em>r</em>-uniform suspension of <em>F</em> is the <em>r</em>-uniform hypergraph obtained from <em>F</em> by taking <span><math><mi>r</mi><mo>−</mo><mn>2</mn></math></span> new vertices and adding them to every edge. In this paper, we consider Turán problems on suspension hypergraphs, and we obtain several general and exact results.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114887"},"PeriodicalIF":0.7,"publicationDate":"2025-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528741","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 : 2025-11-12DOI: 10.1016/j.disc.2025.114883
Jiaqi Liao , Mengyu Cao , Zequn Lv , Mei Lu
Let and . A k-multiset in is a multiset of cardinality k, whose elements are integers from , and each element is allowed to have at most m repetitions. A family of k-multisets in is said to be intersecting if every pair of k-multisets from the family have non-empty intersection. In this paper, we give the size and structure of the largest non-trivial intersecting family of k-multisets in for . In the special case when , our result gives rise to an unbounded multiset version for Hilton-Milner theorem given by Meagher and Purdy. Furthermore, our main theorem unites the statements of the Hilton-Milner theorem for finite sets and unbounded multisets.
{"title":"Hilton-Milner theorem for k-multisets","authors":"Jiaqi Liao , Mengyu Cao , Zequn Lv , Mei Lu","doi":"10.1016/j.disc.2025.114883","DOIUrl":"10.1016/j.disc.2025.114883","url":null,"abstract":"<div><div>Let <span><math><mi>k</mi><mo>,</mo><mi>n</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> and <span><math><mi>m</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span>. A <em>k</em>-multiset in <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mrow><mi>m</mi></mrow></msub></math></span> is a multiset of cardinality <em>k</em>, whose elements are integers from <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>, and each element is allowed to have at most <em>m</em> repetitions. A family of <em>k</em>-multisets in <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mrow><mi>m</mi></mrow></msub></math></span> is said to be intersecting if every pair of <em>k</em>-multisets from the family have non-empty intersection. In this paper, we give the size and structure of the largest non-trivial intersecting family of <em>k</em>-multisets in <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mrow><mi>m</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>⩾</mo><mi>k</mi><mo>+</mo><mrow><mo>⌈</mo><mi>k</mi><mo>/</mo><mi>m</mi><mo>⌉</mo></mrow></math></span>. In the special case when <span><math><mi>m</mi><mo>=</mo><mo>∞</mo></math></span>, our result gives rise to an unbounded multiset version for Hilton-Milner theorem given by Meagher and Purdy. Furthermore, our main theorem unites the statements of the Hilton-Milner theorem for finite sets and unbounded multisets.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114883"},"PeriodicalIF":0.7,"publicationDate":"2025-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528740","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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