Complementary cycles in almost regular multipartite tournaments, where one cycle has length four

Abstract

Let D be a digraph with vertex set V(D) and independence number α(D). If x∈V(D), then the numbers d+(x) and d−(x) are the outdegree and indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D)=max{max(d+(x),d−(x))−min(d+(y),d−(y))∣x,y∈V(D)}. If ig(D)=0, then D is regular, and if ig(D)≤1, then D is almost regular. A c-partite tournament is an orientation of a complete c-partite graph.In 1999, Yeo conjectured that each regular c-partite tournament D with c≥4 and |V(D)|≥8 contains a pair of vertex-disjoint directed cycles of lengths 4 and |V(D)|−4. In 2004, Volkmann confirmed this conjecture for c≥5 and c=4 and α(D)≥4. As a supplement to this result, we prove in this paper the following theorem.Let D be an almost regular c-partite tournament with |V(D)|≥8 such that all partite sets have the same cardinality r. If c≥5 or c=4 and r≥6, then D contains a pair of vertex-disjoint directed cycles of lengths 4 and |V(D)|−4.