{"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}
P. Dankelmann , Y. Guo , E.J. Rivett-Carnac , L. Volkmann
{"title":"The oriented diameter of graphs derived from other graphs","authors":"P. Dankelmann , Y. Guo , E.J. Rivett-Carnac , L. Volkmann","doi":"10.1016/j.disc.2025.114443","DOIUrl":"10.1016/j.disc.2025.114443","url":null,"abstract":"<div><div>The diameter of a strong digraph or connected graph is the largest of the distances between its vertices. An orientation of an undirected graph <em>G</em> is a digraph obtained from <em>G</em> by assigning a direction to each edge. An orientation is said to be strong if the digraph is strongly connected. The oriented diameter of a graph <em>G</em> is the minimum diameter amongst all strong orientations of <em>G</em>. In this paper we give bounds on the oriented diameter of two graphs derived from a given graph: the complement and the line graph. We give bounds on the oriented diameter of the complement of a graph <em>G</em> in terms of the diameter of <em>G</em> and in terms of the oriented diameter of <em>G</em>. As a corollary, we obtain Nordhaus-Gaddum type results for the oriented diameter. We prove that the oriented diameter of the line graph of a graph <em>G</em> cannot exceed the oriented diameter of <em>G</em> by more than 1, and that it is at least, approximately, the square root of the oriented diameter of <em>G</em>. We show that both results, in some sense, are best possible.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114443"},"PeriodicalIF":0.7,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143428032","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":"A characterization on (g,f)-parity orientations","authors":"Xinxin Ma, Hongliang Lu","doi":"10.1016/j.disc.2025.114440","DOIUrl":"10.1016/j.disc.2025.114440","url":null,"abstract":"<div><div>Let <em>G</em> be a graph and <span><math><mi>g</mi><mo>,</mo><mi>f</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>N</mi></mrow></msup></math></span> be two functions such that <span><math><mi>g</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>≤</mo><mi>f</mi><mo>(</mo><mi>v</mi><mo>)</mo><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>g</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>≡</mo><mi>f</mi><mo>(</mo><mi>v</mi><mo>)</mo><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>2</mn><mo>)</mo><mspace></mspace><mtext>for every</mtext><mspace></mspace><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. An orientation <em>O</em> of <em>G</em> is called a <span><math><mo>(</mo><mi>g</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span>-parity orientation if <span><math><mi>g</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>≤</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>O</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>v</mi><mo>)</mo><mo>≤</mo><mi>f</mi><mo>(</mo><mi>v</mi><mo>)</mo></math></span> and <span><math><mi>g</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>≡</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>O</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>v</mi><mo>)</mo><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>2</mn><mo>)</mo></math></span> for every <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In this paper, we give a Tutte-type characterization for a graph to have a <span><math><mo>(</mo><mi>g</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span>-parity orientation.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114440"},"PeriodicalIF":0.7,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143428031","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":"An extension of Pólya's enumeration theorem","authors":"Xiongfeng Zhan, Xueyi Huang","doi":"10.1016/j.disc.2025.114445","DOIUrl":"10.1016/j.disc.2025.114445","url":null,"abstract":"<div><div>In combinatorics, Pólya's Enumeration Theorem is a powerful tool for solving a wide range of counting problems, including the enumeration of groups, graphs, and chemical compounds. In this paper, we present an extension of Pólya's Enumeration Theorem. As an application, we derive a formula that expresses the <em>n</em>-th elementary symmetric polynomial in <em>m</em> indeterminates (where <span><math><mi>n</mi><mo>≤</mo><mi>m</mi></math></span>) as a variant of the cycle index polynomial of the symmetric group <span><math><mrow><mi>Sym</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. This result resolves a problem posed by Amdeberhan in 2012.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114445"},"PeriodicalIF":0.7,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143420447","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":"A combinatorial proof of a family of truncated identities for the partition function","authors":"Yongqiang Chen, Olivia X.M. Yao","doi":"10.1016/j.disc.2025.114434","DOIUrl":"10.1016/j.disc.2025.114434","url":null,"abstract":"<div><div>In 2012, Andrews and Merca proved a truncated partition identity by studying the truncated series of Euler's pentagonal number theorem. Andrews and Merca's work has opened up a new study on truncated theta series and a number of results on truncated theta series have been proved in the past decade. Recently, Xia, Yee and Zhao proved a new truncated partition identity by taking different truncated series than the one chosen by Andrews and Merca. Very recently, Yao proved a new truncated identity on Euler's pentagonal number theorem. The identity is equivalent to a family of truncated identities for the partition function which involves the results proved by Andrew-Merca, and Xia-Yee-Zhao. In this paper, we provide a purely combinatorial proof of the family of truncated identities for the partition function. In particular, we answer a question on combinatorial proofs of two partition identities, which were posed by Wang and Xiao.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114434"},"PeriodicalIF":0.7,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419450","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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