{"title":"Weak submodularity implies localizability: Local search for constrained non-submodular function maximization","authors":"Majun Shi , Qingyong Zhu , Bei Liu , Yuchao Li","doi":"10.1016/j.disc.2024.114287","DOIUrl":"10.1016/j.disc.2024.114287","url":null,"abstract":"<div><div>Local search algorithms are commonly employed to address a variety of problems in the domain of operations research and combinatorial optimization. Most of the literature on the maximization of constrained monotone non-submodular functions is based on a greedy strategy, and few designs of local search approach are involved. In this paper, we explore the problem of maximizing a monotone non-submodular function under a <em>p</em>-matroid (<span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>) constraint with local search algorithms. And we indicate that weak submodularity implies localizability of set function optimization which can be used to offer provable approximation guarantees of local search algorithms.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114287"},"PeriodicalIF":0.7,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534358","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":"Localized version of hypergraph Erdős-Gallai Theorem","authors":"Kai Zhao, Xiao-Dong Zhang","doi":"10.1016/j.disc.2024.114293","DOIUrl":"10.1016/j.disc.2024.114293","url":null,"abstract":"<div><div>The weight function of an edge in an <em>n</em>-vertex uniform hypergraph <span><math><mi>H</mi></math></span> is defined with respect to the number of edges in the longest Berge path containing the edge. We prove that the summation of the weight function values for all edges in <span><math><mi>H</mi></math></span> is at most <em>n</em>, and characterize all extremal hypergraphs that attain this bound. This result strengthens and extends the hypergraph version of the classic Erdős-Gallai Theorem, providing a local version of this theorem.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114293"},"PeriodicalIF":0.7,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142528661","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 note proving the nullity of block graphs is unbounded","authors":"Michael Cary","doi":"10.1016/j.disc.2024.114289","DOIUrl":"10.1016/j.disc.2024.114289","url":null,"abstract":"<div><div>Block graphs are important baseline structures for a vast array of community detection and other network partitioning models. Singular graphs have many important uses in the physical sciences. A recent conjecture was posited that the nullity of a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-free block graph cannot be larger than one. In this paper we prove that the conjecture is false by constructing a family of counterexamples using the Cauchy interlacing theorem for real symmetric matrices. In doing so, we prove the stronger statement that the nullity of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-free block graphs is unbounded. Finally, the implications of this result for the computational network theory literature are discussed.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114289"},"PeriodicalIF":0.7,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445873","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":"Bipartite Ramsey number pairs that involve combinations of cycles and odd paths","authors":"Ernst J. Joubert","doi":"10.1016/j.disc.2024.114283","DOIUrl":"10.1016/j.disc.2024.114283","url":null,"abstract":"<div><div>For bipartite graphs <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, the bipartite Ramsey number <span><math><mi>b</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> is the least positive integer <em>b</em>, so that any coloring of the edges of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>b</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> with <em>k</em> colors, will result in a copy of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> in the <em>i</em>th color, for some <em>i</em>. For bipartite graphs <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, the bipartite Ramsey number pair <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>, denoted by <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>, is an ordered pair of integers such that for any blue-red coloring of the edges of <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span>, with <span><math><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≥</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, either a blue copy of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> exists or a red copy of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> exists if and only if <span><math><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≥</mo><mi>a</mi></math></span> and <span><math><msup><mrow><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≥</mo><mi>b</mi></math></span>. In <span><span>[4]</span></span>, Faudree and Schelp considered bipartite Ramsey number pairs involving paths. Recently, Joubert, Hattingh and Henning showed, in <span><span>[7]</span></span> and <span><span>[8]</span></span>, that <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><mn>2</mn><mi>s</mi><mo>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114283"},"PeriodicalIF":0.7,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441529","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 method for constructing graphs with the same resistance spectrum","authors":"Si-Ao Xu, Huan Zhou, Xiang-Feng Pan","doi":"10.1016/j.disc.2024.114284","DOIUrl":"10.1016/j.disc.2024.114284","url":null,"abstract":"<div><div>Let <em>G</em> be a simple graph with vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and edge set <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. The resistance distance <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> between two vertices <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span> of <em>G</em>, is defined to be the effective resistance between the two vertices in the corresponding electrical network in which each edge of <em>G</em> is replaced by a unit resistor. The resistance spectrum <span><math><mi>RS</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> is the multiset of the resistance distances between all pairs of vertices in the graph. This paper presents a novel method for constructing graphs with the same resistance spectrum. It is obtained that for any positive integer <em>k</em>, there exist at least <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span> graphs with the same resistance spectrum. Furthermore, it is shown that for <span><math><mi>n</mi><mo>≥</mo><mn>10</mn></math></span>, there are at least <span><math><mn>2</mn><mo>(</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>10</mn><mo>)</mo><mi>p</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo><mo>+</mo><mi>q</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo><mo>)</mo></math></span> pairs of graphs of order <em>n</em> with the same resistance spectrum, where <span><math><mi>p</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo></math></span> and <span><math><mi>q</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo></math></span> are the numbers of partitions of the integer <span><math><mi>n</mi><mo>−</mo><mn>9</mn></math></span> and simple graphs of order <span><math><mi>n</mi><mo>−</mo><mn>9</mn></math></span>, respectively.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114284"},"PeriodicalIF":0.7,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434201","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":"Star-critical Ramsey numbers involving large books","authors":"Xun Chen , Qizhong Lin , Lin Niu","doi":"10.1016/j.disc.2024.114270","DOIUrl":"10.1016/j.disc.2024.114270","url":null,"abstract":"<div><div>For graphs <span><math><mi>F</mi><mo>,</mo><mi>G</mi></math></span> and <em>H</em>, let <span><math><mi>F</mi><mo>→</mo><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> signify that any red/blue edge coloring of <em>F</em> contains either a red <em>G</em> or a blue <em>H</em>. The Ramsey number <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is defined to be the smallest integer <em>r</em> such that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>→</mo><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>. Let <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup></math></span> be the book graph which consists of <em>n</em> copies of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> all sharing a common <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, and let <span><math><mi>G</mi><mo>:</mo><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span> be the complete <span><math><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-partite graph with <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span>, <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>.</div><div>In this paper, avoiding the use of Szemerédi's regularity lemma, we show that for any fixed <span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>, <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and sufficiently large <em>n</em>, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>(</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></mrow></msub><mo>∖</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>2</mn></mrow></msub><mo>→</mo><mo>(</mo><mi>G</mi><mo>,</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>)</mo></math></span>. This implies that the star-critical Ramsey number <span><math><msub><mrow><mi>r</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><msubsup><mrow><mi>B</mi></mrow><m","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114270"},"PeriodicalIF":0.7,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426770","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":"The Mostar index of Tribonacci cubes","authors":"Yu Wang, Min Niu","doi":"10.1016/j.disc.2024.114281","DOIUrl":"10.1016/j.disc.2024.114281","url":null,"abstract":"<div><div>Tribonacci cubes <span><math><msubsup><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> are a class of hypercube-like cubes obtained by removing all vertices of hypercubes <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> that have more than two consecutive 1s. In this paper, we calculate the Mostar index of Tribonacci cubes, which is a measure of how far the graph is from being distance-balanced and is used to study various properties of chemical graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114281"},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426771","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":"Structural parameters of Schnyder woods","authors":"Christian Ortlieb , Jens M. Schmidt","doi":"10.1016/j.disc.2024.114282","DOIUrl":"10.1016/j.disc.2024.114282","url":null,"abstract":"<div><div>We study two fundamental parameters of Schnyder woods by exploiting structurally related methods. First, we prove a new lower bound on the total number of leaves in the three trees of a Schnyder wood. Second, it is well-known that Schnyder woods can be used to find three compatible ordered path partitions. We prove new lower bounds on the number of singletons, i.e. paths that consists of exactly one vertex, in such compatible ordered path partitions. All bounds that we present are tight.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114282"},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426769","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":"On isometry and equivalence of skew constacyclic codes","authors":"Hassan Ou-azzou , Mustapha Najmeddine , Nuh Aydin","doi":"10.1016/j.disc.2024.114279","DOIUrl":"10.1016/j.disc.2024.114279","url":null,"abstract":"<div><div>In this paper we generalize the notion of <em>n</em>-isometry and <em>n</em>-equivalence relation introduced by Chen et al. in <span><span>[13]</span></span>, <span><span>[12]</span></span> to classify constacyclic codes of length <em>n</em> over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>q</mi></mrow></msub></mrow></msub></math></span>, where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> is a prime power, to the case of skew constacyclic codes without derivation. We call these relations respectively <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-equivalence and <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-isometric relation, where <em>n</em> is the length of the code and <em>σ</em> is an automorphism of the finite field. We compute the number of <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-equivalence and <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-isometric classes, and we give conditions on <em>λ</em> and <em>μ</em> for which <span><math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span>-constacyclic codes and <span><math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></math></span>-constacyclic codes are equivalent. Under some conditions on <em>n</em> and <em>q</em> we prove that skew constacyclic codes are equivalent to cyclic codes by using properties of our equivalence relation introduced. We also prove that when <em>q</em> is even and <em>σ</em> is the Frobenius automorphism, skew constacyclic codes of length <em>n</em> are equivalent to cyclic codes when <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Finally we give some examples as applications of the theory developed here.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114279"},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142416804","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 cyclic symmetric Hamilton cycle decompositions of complete multipartite graphs","authors":"Fatima Akinola , Michael W. Schroeder","doi":"10.1016/j.disc.2024.114277","DOIUrl":"10.1016/j.disc.2024.114277","url":null,"abstract":"<div><div>A decomposition of a graph with <em>n</em> vertices, labeled by <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, is cyclic if addition by 1 to the vertices acts on the decomposition, and the decomposition is <em>d</em>-symmetric for a divisor <em>d</em> of <em>n</em> if addition by <span><math><mi>n</mi><mo>/</mo><mi>d</mi></math></span> to the vertices acts invariantly on the decomposition. In a 2017 paper, Merola et al. established the necessary and sufficient conditions under which a complete multipartite graph with an even number of parts, each with <em>d</em> vertices, has a cyclic Hamilton cycle decomposition; these decompositions were also <em>d</em>-symmetric.</div><div>In this paper we establish the necessary and sufficient conditions for the analogous question with complete multipartite graphs with an odd number of parts, which settles the existence of cyclic, <em>d</em>-symmetric Hamilton cycle decompositions for all balanced, complete multipartite graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114277"},"PeriodicalIF":0.7,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142416657","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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