{"title":"Poset positional games","authors":"Guillaume Bagan , Eric Duchêne , Florian Galliot , Valentin Gledel , Mirjana Mikalački , Nacim Oijid , Aline Parreau , Miloš Stojaković","doi":"10.1016/j.disc.2025.114455","DOIUrl":"10.1016/j.disc.2025.114455","url":null,"abstract":"<div><div>We propose a generalization of positional games, supplementing them with a restriction on the order in which the elements of the board are allowed to be claimed. We introduce poset positional games, which are positional games with an additional structure – a poset on the elements of the board. Throughout the game play, based on this poset and the set of the board elements that are claimed up to that point, we reduce the set of available moves for the player whose turn it is – an element of the board can only be claimed if all the smaller elements in the poset are already claimed.</div><div>We proceed to analyze these games in more detail, with a prime focus on the most studied convention, the Maker-Breaker games. First we build a general framework around poset positional games. Then, we perform a comprehensive study of the complexity of determining the game outcome, conditioned on the structure of the family of winning sets on the one side and the structure of the poset on the other.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114455"},"PeriodicalIF":0.7,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479707","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":"The signless Laplacian spectral radius of book-free graphs","authors":"Ming-Zhu Chen , Ya-Lei Jin , Peng-Li Zhang","doi":"10.1016/j.disc.2025.114448","DOIUrl":"10.1016/j.disc.2025.114448","url":null,"abstract":"<div><div>Given a graph <em>H</em>, a graph is called <em>H</em>-free if it does not contain <em>H</em> as a subgraph. For a positive integer <em>k</em>, a book <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is a graph consisting of <span><math><mi>k</mi><mo>+</mo><mn>1</mn></math></span> triangles sharing a common edge. In this paper, let <em>G</em> be a <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free graph of order <span><math><mi>n</mi><mo>≥</mo><mn>49</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>22</mn><mi>k</mi><mo>+</mo><mn>4</mn></math></span>. Then the signless Laplacian spectral radius <span><math><mi>q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><msqrt><mrow><msup><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mi>k</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>8</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> with equality if and only if <span><math><mi>G</mi><mo>=</mo><msub><mrow><mover><mrow><mi>K</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>∨</mo><mi>H</mi></math></span>, where <em>k</em> or <span><math><mi>n</mi><mo>−</mo><mi>k</mi></math></span> is even, and <em>H</em> is a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-free <em>k</em>-regular graph of order <span><math><mi>n</mi><mo>−</mo><mi>k</mi></math></span>. Furthermore, if <em>k</em> and <span><math><mi>n</mi><mo>−</mo><mi>k</mi></math></span> are both odd, the extremal graphs with the maximum signless Laplacian spectral radius are also characterized.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114448"},"PeriodicalIF":0.7,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143452754","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":"Hoffmann-Ostenhof's 3-Decomposition Conjecture","authors":"Genghua Fan, Chuixiang Zhou","doi":"10.1016/j.disc.2025.114454","DOIUrl":"10.1016/j.disc.2025.114454","url":null,"abstract":"<div><div>The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a set of cycles, and a matching. It has been proved independently by different groups of people that every connected cubic graph can be decomposed into a spanning tree, a set of cycles, and a set of vertex-disjoint paths of at most two edges. In this paper, we establish a bound on the number of paths of two edges, proving that every connected cubic graph on <em>n</em> vertices can be decomposed into a spanning tree, a set of cycles, and a set of vertex-disjoint paths of at most two edges such that the number of paths of two edges is at most <span><math><mfrac><mrow><mi>n</mi><mo>−</mo><mn>4</mn></mrow><mrow><mn>6</mn></mrow></mfrac></math></span>. Our proof is based on a structural analysis, which might provide a new approach to attack the 3-Decomposition Conjecture.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114454"},"PeriodicalIF":0.7,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143444737","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}
Michael D. Barrus , Ann N. Trenk , Rebecca Whitman
{"title":"The hereditary closure of the unigraphs","authors":"Michael D. Barrus , Ann N. Trenk , Rebecca Whitman","doi":"10.1016/j.disc.2025.114444","DOIUrl":"10.1016/j.disc.2025.114444","url":null,"abstract":"<div><div>A graph with degree sequence <em>π</em> is a <em>unigraph</em> if it is isomorphic to every graph that has degree sequence <em>π</em>. The class of unigraphs is not hereditary and in this paper we study the related hereditary class HCU, the hereditary closure of unigraphs, consisting of all graphs induced in a unigraph. We characterize the class HCU in multiple ways making use of the tools of a decomposition due to Tyshkevich. We also provide a new characterization of the class that consists of unigraphs for which all induced subgraphs are also unigraphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114444"},"PeriodicalIF":0.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437956","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":"Some online Maker-Breaker games","authors":"Patrick Bennett , Alan Frieze","doi":"10.1016/j.disc.2025.114446","DOIUrl":"10.1016/j.disc.2025.114446","url":null,"abstract":"<div><div>We consider some Maker-Breaker games of the following flavor. We have some set <em>V</em> of items for purchase. Maker's goal is to purchase some member of a given family <span><math><mi>H</mi></math></span> of subsets of <em>V</em> as cheaply as possible and Breaker's goal is to make the purchase as expensive as possible. Each player has a pointer and during a player's turn their pointer moves through the items in the order of the permutation until the player decides to take one. We mostly focus on the case where the permutation is random and unknown to the players (it is revealed by the players as their pointers move).</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114446"},"PeriodicalIF":0.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437939","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":"Minimum degree k and k-connectedness usually arrive together","authors":"Sahar Diskin , Anna Geisler","doi":"10.1016/j.disc.2025.114453","DOIUrl":"10.1016/j.disc.2025.114453","url":null,"abstract":"<div><div>Let <span><math><mi>d</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></math></span> be such that <span><math><mi>d</mi><mo>=</mo><mi>ω</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>, and <span><math><mi>d</mi><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>a</mi></mrow></msup></math></span> for some constant <span><math><mi>a</mi><mo>></mo><mn>0</mn></math></span>. Consider a <em>d</em>-regular graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> and the random graph process that starts with the empty graph <span><math><mi>G</mi><mo>(</mo><mn>0</mn><mo>)</mo></math></span> and at each step <span><math><mi>G</mi><mo>(</mo><mi>i</mi><mo>)</mo></math></span> is obtained from <span><math><mi>G</mi><mo>(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> by adding uniformly at random a new edge from <em>E</em>. We show that if <em>G</em> satisfies some (very) mild global edge-expansion, and an almost optimal edge-expansion of sets up to order <span><math><mi>O</mi><mo>(</mo><mi>d</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>, then for any constant <span><math><mi>k</mi><mo>∈</mo><mi>N</mi></math></span> in the random graph process on <em>G</em>, typically the hitting times of minimum degree at least <em>k</em> and of <em>k</em>-connectedness are equal. This, in particular, covers both <em>d</em>-regular high dimensional product graphs and pseudo-random graphs, and confirms a conjecture of Joos from 2015. We further demonstrate that this result is tight in the sense that there are <em>d</em>-regular <em>n</em>-vertex graphs with optimal edge-expansion of sets up to order <span><math><mi>Ω</mi><mo>(</mo><mi>d</mi><mo>)</mo></math></span>, for which the probability threshold of minimum degree at least one is different than the probability threshold of connectivity.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114453"},"PeriodicalIF":0.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445239","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":"Csikvári's poset and Tutte polynomial","authors":"Changxin Ding","doi":"10.1016/j.disc.2025.114450","DOIUrl":"10.1016/j.disc.2025.114450","url":null,"abstract":"<div><div>Csikvári constructed a poset on trees to prove that several graph functions attain extreme values at the star and the path among the trees on a fixed number of vertices. Reiner and Smith proved that the Tutte polynomials <span><math><mi>T</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>)</mo></math></span> of cones over trees, which are the graphs obtained by attaching a cone vertex to a tree, have the described extreme behavior. They further conjectured that the result can be strengthened in terms of Csikvári's poset. We solve this conjecture affirmatively.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114450"},"PeriodicalIF":0.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143444331","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}
Marién Abreu , Giuseppe Mazzuoccolo , Federico Romaniello , Jean Paul Zerafa
{"title":"The Pairing-Hamiltonian property in graph prisms","authors":"Marién Abreu , Giuseppe Mazzuoccolo , Federico Romaniello , Jean Paul Zerafa","doi":"10.1016/j.disc.2025.114441","DOIUrl":"10.1016/j.disc.2025.114441","url":null,"abstract":"<div><div>Let <em>G</em> be a graph of even order, and consider <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> as the complete graph on the same vertex set as <em>G</em>. A perfect matching of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> is called a pairing of <em>G</em>. If for every pairing <em>M</em> of <em>G</em> it is possible to find a perfect matching <em>N</em> of <em>G</em> such that <span><math><mi>M</mi><mo>∪</mo><mi>N</mi></math></span> is a Hamiltonian cycle of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>, then <em>G</em> is said to have the Pairing-Hamiltonian property, or PH-property, for short. In 2007, Fink (2007) <span><span>[4]</span></span> proved that for every <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, the <em>d</em>-dimensional hypercube <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> has the PH-property, thus proving a conjecture posed by Kreweras in 1996. In this paper we extend Fink's result by proving that given a graph <em>G</em> having the PH-property, the prism graph <span><math><mi>P</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>G</mi><mo>□</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of <em>G</em> has the PH-property as well. Moreover, if <em>G</em> is a connected graph, we show that there exists a positive integer <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> such that the <span><math><msup><mrow><mi>k</mi></mrow><mrow><mtext>th</mtext></mrow></msup></math></span>-prism of a graph <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> has the PH-property for all <span><math><mi>k</mi><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114441"},"PeriodicalIF":0.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437938","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}
Maximilian Gorsky , Theresa Johanni , Sebastian Wiederrecht
{"title":"A note on the 2-factor Hamiltonicity Conjecture","authors":"Maximilian Gorsky , Theresa Johanni , Sebastian Wiederrecht","doi":"10.1016/j.disc.2025.114442","DOIUrl":"10.1016/j.disc.2025.114442","url":null,"abstract":"<div><div>The 2-factor Hamiltonicity Conjecture by Funk, Jackson, Labbate, and Sheehan [JCTB, 2003] asserts that all cubic, bipartite graphs in which all 2-factors are Hamiltonian cycles can be built using a simple operation starting from <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span> and the Heawood graph.</div><div>We discuss the link between this conjecture and matching theory, in particular by showing that this conjecture is equivalent to the statement that the two exceptional graphs in the conjecture are the only cubic braces in which all 2-factors are Hamiltonian cycles, where braces are connected, bipartite graphs in which every matching of size at most two is contained in a perfect matching. In the context of matching theory this conjecture is especially noteworthy as <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span> and the Heawood graph are both strongly tied to the important class of Pfaffian graphs, with <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span> being the canonical non-Pfaffian graph and the Heawood graph being one of the most noteworthy Pfaffian graphs.</div><div>Our main contribution is a proof that the Heawood graph is the only Pfaffian, cubic brace in which all 2-factors are Hamiltonian cycles. This is shown by establishing that, aside from the Heawood graph, all Pfaffian braces contain a cycle of length four, which may be of independent interest.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114442"},"PeriodicalIF":0.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437954","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":"Jacobi polynomials, invariant rings, and generalized t-designs","authors":"Himadri Shekhar Chakraborty , Nur Hamid , Tsuyoshi Miezaki , Manabu Oura","doi":"10.1016/j.disc.2025.114447","DOIUrl":"10.1016/j.disc.2025.114447","url":null,"abstract":"<div><div>In the present paper, we provide results that relate the Jacobi polynomials in genus <em>g</em>. We show that if a code is <em>t</em>-homogeneous that is, the codewords of the code for every given weight hold a <em>t</em>-design, then its Jacobi polynomial in genus <em>g</em> with composition <em>T</em> with <span><math><mo>|</mo><mi>T</mi><mo>|</mo><mo>≤</mo><mi>t</mi></math></span> can be obtained from its weight enumerator in genus <em>g</em> using the polarization operator. Using this fact, we investigate the invariant ring, which relates the homogeneous Jacobi polynomials of the binary codes in genus <em>g</em>. Specifically, the generators of the invariant ring appearing for <span><math><mi>g</mi><mo>=</mo><mn>1</mn></math></span> are obtained. Moreover, we define the split Jacobi polynomials in genus <em>g</em> and obtain the MacWilliams type identity for it. A split generalization for higher genus cases of the relation between the Jacobi polynomials and weight enumerator of a <em>t</em>-homogeneous code also given.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114447"},"PeriodicalIF":0.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437955","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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