The 1-factors of Oriented Graphs

1. Introduction. A graph G consists of a set 7 of elements called vertices together with a set E of elements called edges, the two sets having no element in common. With each edge there are associated either one or two vertices called its ends. In this paper we consider only graphs in which each edge has two distinct ends. We call such a graph oriented if one end of each edge is distinguished as the positive and the other as the negative end. If x is the positive end and y the negative end of an edge A we say that x is joined to y by A or that y is joined from x by A. We say also that A is a join from x to y. An oriented graph G is finite if both 7 and £ are finite, and is infinite otherwise. If G is oriented and xEVwe denote by d\x) and d"ix) the cardinals of, the sets of edges of G having x as positive and negative end respectively. We call G simple if d'ix)=d"ix) for each xEV. An oriented graph is regular of degree d if d\x) = d"ix) ==d for each xE V. An oriented graph 77 is a subgraph of the oriented graph G if the vertices and edges of 77 are vertices and edges respectively of G and if each edge of 77 has the same positive end and the same negative end in 77 as in G.1 If in addition each vertex of G is a vertex of 77, then 77 is a spanning subgraph of G. If 5Ç 7 there is a subgraph of G whose vertices are the members of 5 and whose edges are those members of E which have both their ends in 5. We denote this sub-graph by G[S]. We find it convenient to write Gs for the graph