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.
在有顶点集 V 的图 G 中，一个联盟由两个不相交的集 V 1、V 2 ⊂ V 组成，使得 V 1 和 V 2 都不是支配集，但联盟 V 1 ∪ V 2 是 G 中的一个支配集。如果 π 的每个非支配集都是一个联盟的成员，且每个支配集都是单顶点集，则 V 的一个分区称为联盟分区 π。每个联盟分区都会生成联盟图。联盟图的顶点与分区集一一对应，当且仅当两个顶点对应的集构成一个联盟时，这两个顶点相邻。在论文 [T. W. Haynes, J. T.W. Haynes, J. T. Hedetniemi, S. T. Hedetniemi, A. A. McRae, R. Mohan, Discuss.Math.Graph Theory 43 (2023) 931-946]，作者证明了循环的分区联盟只能生成 27 个联盟图，并提出了拥有最多联盟图的最短循环的问题。在本文中，我们证明 C 15 是具有这一性质的最短图。
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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 .
二色数和二色数是色数和消色数的广义，适用于考虑非循环着色的数图。在本文中，我们确定了在合适的 m 条件下，由接受完整 k 着色的数图的 Zykov 和所产生的数图的重色数，k = 1+ √ 1+4 m 2。因此，该族中每个数图的对色数等于和谐数。特别是，我们确定了由完整数图的哈密顿因式的齐可夫和在合适数图上产生的数图的重色数。我们还得到了图的等效结果。此外，我们还根据完整图的厚度确定了由广义组合产生的数图的消色数。最后，我们将关于锦标赛的齐可夫和的消色数的一些结果扩展到非锦标赛的数图类别，并将它们和关于消色数的结果应用于在 2 ≤ r ≤ t 的一些特殊情况下存在消色数为 r 和消色数为 t 的数图的问题。
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{"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}
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