Weakly Complementary Cycles in 3-Connected Multipartite Tournaments

• Journal title : Kyungpook mathematical journal
• Volume 48, Issue 2,  2008, pp.287-302
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2008.48.2.287
Title & Authors
Weakly Complementary Cycles in 3-Connected Multipartite Tournaments
Volkmann, Lutz; Winzen, Stefan;

Abstract
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 $\small{C_1}$ and $\small{C_2}$ such that V(D) = $\small{V(C_1)\;{\cup}\;V(C_2)}$, and a multipartite tournament D is called weakly cycle complementary if there exist two vertex disjoint cycles $\small{C_1}$ and $\small{C_2}$ such that $\small{V(C_1)\;{\cup}\;V(C_2)}$ 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 $\small{{\mid}V(T)\mid}$ - t for all $\small{3\;{\leq}\;t\;{\leq}\;{\mid}V(T)\mid/2}$. Recently, Volkmann [8] proved that each regular multipartite tournament D of order $\small{{\mid}V(D)\mid\;\geq\;8}$ 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 $\small{c\;\geq\;3}$ that are weakly cycle complementary.
Keywords
Multipartite tournaments;weakly cycle complementarity;
Language
English
Cited by
1.
Complementary Cycles in Irregular Multipartite Tournaments, Mathematical Problems in Engineering, 2016, 2016, 1
2.
Weakly Cycle Complementary 3-Partite Tournaments, Graphs and Combinatorics, 2011, 27, 5, 669
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