On cycles through a given vertex in multipartite tournaments
An \(n\)-partite tournament is a directed graph obtained from a complete \(n\)-partite graph by assigning directions to its edges. The authors show that if \(D\) is a strongly connected \(n\)-partite tournament with partite sets \(V_1,V_2,\dots, V_n\) and \(v\) is an arbitrary vertex of \(D\), then (1) \(v\) belongs to some longest cycle of \(D\), (2) if \(n\geq 3\), then \(v\) belongs to an \(m\)- or \((m+1)\)-cycle for every \(m\in \{3,4,\dots,n\}\), (3) if \(n\geq 4\) and \(|V_i|\geq 2\) for \(i=1,2,\dots,n\), then \(v\) belongs to an \((n+1)\)- or \((n+2)\)-cycle of \(D\). Known results due to Bondy, Ayel and Gutin are then deduced from these results.
scientific article; zbMATH DE number 1019059