Long paths and cycles in oriented graphs

We obtain several sufficient conditions on the degrees of an oriented graph for the existence of long paths and cycles. As corollaries of our results we deduce that a regular tournament contains an edge-disjoint Hamilton cycle and path, and that a regular bipartite tournament is hamiltonian. An orientedgraph is a directed simple graph, that is to say, a digraph without loops, mutiple arcs, or cycles of length two. Many authors have obtained various degree conditions which imply that certain families of graphs, or digraphs, contain long paths and cycles. The corresponding literature for oriented graphs, however, is concerned almost entirely with tournaments, which are oriented complete graphs. In this paper we consider the problem for other families of oriented graphs. We first work with oriented graphs of fixed minimum in-degree, but an arbitrary number of vertices, and then specialise by also fixing the minimum out-degree. We next obtain some hamiltonian conditions by demanding that the total number of vertices be relatively small. Finally, we prove a result concerning the lengths of cycles in oriented complete bipartite graphs. We give many conjectures throughout, which indicate that the majority of our results are far from being best possible. All terms not explicitly defined in this paper may be found in [4]. We note, however, that we shall refer to spanning paths, and spanning cycles, as Hamilton paths, and Hamilton cycles, respectively. Let R be an oriented graph and S be a sugraph of R. Denote the set of vertices and arcs of S by V(S) andA(S), respectively. For Y E V(R) and B S V(R), define Nj(u) and G ( v ) to be the set of vertices of B which, respectively, dominate, and are dominated by, the vertex Y. Put To simplify notation, we shall denote N&,(v), N&,(v), and NQS)(Y) by &(v), N&), and N&). We shall refer to I NR(u) I , 1 N ~ ( Y ) 1 , and Journal of Graph Theory, Vol. 5 ( 1 981 ) 145-1 57 o 1981 by John Wiley &Sons, Inc. CCC 0364-9024/81/020145-I 3$01.30 146 JOURNAL OF GRAPH THEORY 1 NR(v) / as the in-degree, the out-degree, and the degree, of v in R and denote them by &(v), dRf(v), and d(v) , respectively. The oriented graph R is said to be kdiregular, or more simply diregular, if &(v) = k = d,f(v ) for all v E V(R): and mdisconnected if, for any distinct pair of vertices u, v E V(R), there exist m internally disjoint paths from u to v . If R is 1diconnected, we shall say simply that R is diconnected. Lemma 1. Let P = v l v 1 *vn be a longest path in an oriented graph R . If d i ( v I ) 2 1, then R contains a cycle of length at least d i ( v , ) -I2. Proof. Since P is a longest path in R,&(vl) C V(P). L e t j = max (i 1 v 1 E M U I N and Put Then ( v l , v2) U N i ( v l ) C V(C). Moreover, since R is an oriented graph, { v I , v2) n &(vl ) = 0, and hence Lemma 2. Every oriented graph of minimum in-degree k , contains a cycle of length at least k + 2. Roo$ Immediate from Lemma 1. Lemma 2 is, in a sense, best possible since there exist oriented graphs of minimum in-degree k, which contain no cycles of length greater thank -I2. To illustrate this we shall construct, recursively, a family of graphs, (RkJkZl , with the following properties. (1) Rk is diconnected. (2) Rk has a distinguished vertex v, such that Gk(uk) = k -I1, and &,(v) = k for all Y E V(R,)\ (v,).