Vertex disjoint cycles in a directed graph

Let D be a directed graph of order n 4 and minimum degree at least (3n 3)/2. Let n = nI + n2 where nl 2 and n2 2. Then D contains two vertex-disjoint directed cycles of nI and n2 respectively. The result is sharp if n ~ 6: we give counterexamples if the condition on the minimum degree is relaxed. 1 Introduction We discuss only finite simple graphs and strict and use standard terminology and notation from [3] except as indicated. In 1963, Corradi and Hajnal [4J the maximum number of vertex-disjoint cycles in a graph. They proved that if G is a graph of order at least 3k with minimum degree at least 2k, then G contains k vertex-disjoint In particular, when the order of G is exactly 3k, then G contains k vertex-disjoint triangles. In 1984 EI-Zahar [5] proved that if G is a graph of order n = nI +n2 with ni 3, i = 1,2 and minimum degree at least fnI/21 + fn2/21, then G contains two vertex-disjoint cycles of lengths nI, n2, respectively. In 1991, Amar and Raspaud [1] investigated vertex-disjoint dicycles in a strongly connected digraph of order n with (n 1)(n-2) + 3 arcs. In this paper, we discuss two vertex-disjoint dicycles in a digraph, proving the following result and showing that it is sharp for all n ;:::: 6. Let D be a digraph of order n ;:::: 4 such that the minimum degree of D is at least (3n 3)/2. Then D contains two vertex-disjoint dicycles of lengths nl and n2) respectively! for any integer partition n nl + n2 with nI ~ 2 and n2 ~ 2. To prove our result, we recall some terminology and notation. Let G be a graph and D a digraph. We use V(G) and E(G) to denote the vertex set and the edge set respectively, of G. We use V(D) and E(D) to denote the vertex set and arc set respectively, of D. A similar notation is used for the vertex sets and edge sets or arc sets of paths and cycles. The degree dG(x) or dD(x) of a vertex x in G or D respectively is the number of edges or arcs incident on it. We use 5(G) and 5(D) for the minimum degree of a vertex in G or D respectively.