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The shortest cycle having the maximal number of coalition graphs 联盟图数量最多的最短周期
Discrete Mathematics Letters Pub Date : 2024-07-17 DOI: 10.47443/dml.2024.111
Andrey A. Dobrynin, H. Golmohammadi
{"title":"The shortest cycle having the maximal number of coalition graphs","authors":"Andrey A. Dobrynin, H. Golmohammadi","doi":"10.47443/dml.2024.111","DOIUrl":"https://doi.org/10.47443/dml.2024.111","url":null,"abstract":"A coalition in a graph G with a vertex set V consists of two disjoint sets V 1 , V 2 ⊂ V , such that neither V 1 nor V 2 is a dominating set, but the union V 1 ∪ V 2 is a dominating set in G . A partition of V is called a coalition partition π if every non-dominating set of π is a member of a coalition and every dominating set is a single-vertex set. Every coalition partition generates its coalition graph. The vertices of the coalition graph correspond one-to-one with the partition sets and two vertices are adjacent if and only if their corresponding sets form a coalition. In the paper [T. W. Haynes, J. T. Hedetniemi, S. T. Hedetniemi, A. A. McRae, R. Mohan, Discuss. Math. Graph Theory 43 (2023) 931–946], the authors proved that partition coalitions of cycles can generate only 27 coalition graphs and asked about the shortest cycle having the maximum number of coalition graphs. In this paper, we show that C 15 is the shortest graph having this property.","PeriodicalId":503566,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141830417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Zykov sums of digraphs with diachromatic number equal to its harmonious number 二色数等于和谐数的数形的齐可夫和
Discrete Mathematics Letters Pub Date : 2024-07-17 DOI: 10.47443/dml.2023.214
M. Olsen, Christian Rubio-Montiel
{"title":"Zykov sums of digraphs with diachromatic number equal to its harmonious number","authors":"M. Olsen, Christian Rubio-Montiel","doi":"10.47443/dml.2023.214","DOIUrl":"https://doi.org/10.47443/dml.2023.214","url":null,"abstract":"The dichromatic number and the diachromatic number are generalizations of the chromatic number and the achromatic number for digraphs considering acyclic colorings. In this paper, we determine the diachromatic number of digraphs arising from the Zykov sum of digraphs that accept a complete k -coloring with k = 1+ √ 1+4 m 2 for a suitable m . As a consequence, the diachromatic number equals the harmonious number for every digraph in this family. In particular, we determine the diachromatic number of digraphs arising from the Zykov sum of Hamiltonian factorizations of complete digraphs over a suitable digraph. We also obtain the equivalent results for graphs. Furthermore, we determine the achromatic number for digraphs arising from the generalized composition in terms of the thickness of complete graphs. Finally, we extend some results on the dichromatic number of Zykov sums of tournaments to the class of digraphs that are not tournaments and apply them, and the results obtained for the diachromatic number, to the problem of the existence of a digraph with dichromatic number r and diachromatic number t for some particular cases with 2 ≤ r ≤ t .","PeriodicalId":503566,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141828324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Inversion sequences and signed permutations 反转序列和有符号排列
Discrete Mathematics Letters Pub Date : 2024-07-17 DOI: 10.47443/dml.2024.060
{"title":"Inversion sequences and signed permutations","authors":"","doi":"10.47443/dml.2024.060","DOIUrl":"https://doi.org/10.47443/dml.2024.060","url":null,"abstract":"","PeriodicalId":503566,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141829152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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