If x is a vertex of a digraph D, then we denote by d + (x) and d − (x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined byIf i g (D) = 0, then D is regular and if i g (D) ≤ 1, then D is called almost regular. A c-partite tournament is an orientation of a complete c-partite graph. Recently, Volkmann and Winzen [L. Volkmann, S. Winzen, Almost regular c-partite tournaments contain a strong subtournament of order c when c ≥ 5, Discrete Math. (2007), 10.1016/j.disc.2006.10.019] showed that every almost regular c-partite tournament D with c ≥ 5 contains a strongly connected subtournament of order p for every p ∈ {3, 4, . . . , c}. In this paper for the class of regular multipartite tournaments we will consider the more difficult question for the existence of strong subtournaments containing a given vertex. We will prove that each vertex of a regular multipartite tournament D with c ≥ 7 partite sets is contained in a strong subtournament of order p for every p ∈ {3, 4, . . . , c − 4}.
A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If x is a vertex of a digraph D, then we denote by d + (x) and d − (x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is deÿned by ig(D) = max{d + (x); d − (x)} − min{d + (y); d − (y)} over all vertices x and y of D (including x = y). If ig(D) = 0, then D is regular and if ig(D) 6 1, then D is called almost regular.Recently, Volkmann and Yeo have proved that every arc of a regular multipartite tournament is contained in a directed Hamiltonian path. If c ¿ 4, then this result remains true for almost all c-partite tournaments D of a given constant irregularity ig(D). For the case that ig(D) = 1 we will give a more detailed analysis. If c = 3, then there exist inÿnite families of such digraphs, which have an arc that is not contained in a directed Hamiltonian path of D. Nevertheless, we will present an interesting su cient condition for an almost regular 3-partite tournament D with the property that a given arc is contained in a Hamiltonian path of D. Terminology and introductionIn this paper all digraphs are ÿnite without loops and multiple arcs. The vertex set and arc set of a digraph D is denoted by V (D) and E(D), respectively. If xy is an arc of a digraph D, then we write x → y and say x dominates y, and if X and Y are two disjoint vertex sets or subdigraphs of D such that every vertex of X dominates every vertex of Y , then we say that X dominates Y , denoted by X → Y . Furthermore, X ❀ Y denotes the fact that there is no arc leading from Y to X . By d(X; Y ) we denote the number of arcs from the set X to the set Y , i.e., d(X; Y ) = |{xy ∈ E(D) : x ∈ X; y ∈ Y }|. If D is a digraph, then the out-neighborhood N + D (x) = N + (x) of a vertex x is the set of vertices dominated by x and the in-neighborhood N − D (x) = N − (x) is the set of vertices dominating x. Therefore, if there is the arc xy ∈ E(D), then y is an outer neighbor of x and x is an inner neighbor of y. The numbers d +| are called the outdegree and indegree of x, respectively. For a vertex set X of D, we deÿne D[X ] as the subdigraph induced by X . If we speak of a cycle or path, then we mean a directed cycle or directed path, and a cycle of length n is called an n-cycle. With x + (x − ) we declare the successor (the predecessor) of a vertex x in a cycle or path. A cycle or path of a digraph D is Hamiltonian, if it includes all the vertices of D. A digraph D is strongly connected or strong, if, for each pair of vertices u and v, there is a path in D from u to v. A digraph D with at least k + 1 vertices is k-strong, if for any set A of at most k − 1 vertices the subdigraph D − A obtained by deleting A is strong. The connectivity of
The vertex set of a digraph D is denoted by V (D). A c-partite tournament is an orientation of a complete c-partite graph. A digraph D is called cycle complementary if there exist two vertex disjoint cycles C1 and C2 such that V (D) = V (C1) ∪ V (C2), and a multipartite tournament D is called weakly cycle complementary if there exist two vertex disjoint cycles C 1 and C 2 such that V (C 1) ∪ V (C 2) contains vertices of all partite sets of D. The problem of complementary cycles in 2-connected tournaments was completely solved by Reid [4] in 1985 and Z. Song [5] in 1993. They proved that every 2-connected tournament T on at least 8 vertices has complementary cycles of length t and |V (T)| − t for all 3 ≤ t ≤ |V (T)|/2. Recently, Volkmann [8] proved that each regular multipartite tournament D of order |V (D)| ≥ 8 is cycle complementary. In this article, we analyze multipartite tournaments that are weakly cycle complementary. Especially, we will characterize all 3-connected c-partite tournaments with c ≥ 3 that are weakly cycle complementary.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.