The Factors of Graphs
نویسنده
چکیده
1. Introduction. A graph G consists of a non-null set V of objects called vertices together with a set E of objects called edges, the two sets having no common element. With each edge there are associated just two vertices, called its ends. Two or more edges may have the same pair of ends. G is finite if both F and E are finite, and infinite otherwise. The degree d G {a) of a vertex a of G is the number of edges of G which have a as an end. G is locally finite if the degree of each vertex of G is finite. Thus the locally finite graphs include the finite graphs as special cases. A subgraph H of G is a graph contained in G. That is, the vertices and edges of H are vertices and edges of G, and an edge of H has the same ends in H as in G. A restriction of G is a subgraph of G which includes all the vertices of G. A graph is said to be regular of order n if the degree of each of its vertices is n. An n-factor of a graph G is a restriction of G which is regular of order n. The problem of finding conditions for the existence of an w-factor of a given graph has been studied by various authors [3; 4; 5]. It has been solved, in part, by Petersen for the case in which the given graph is regular. The author has given a necessary and sufficient condition that a given locally finite graph shall have a 1-factor [6; 7]. In this paper we establish a necessary and sufficient condition that a given locally finite graph shall have an w-factor, where n is any positive integer. Actually we obtain a more general result. We suppose given a function / which associates with each vertex a of a given locally finite graph G a positive integer f(a), and obtain a necessary and sufficient condition that G shall have a restriction H such that d H {a) = f{a) for each vertex a of G. The discussion is based on the method of alternating paths introduced by Petersen [4]. We also consider the problem of associating a non-negative integer with each edge of G so that for each vertex c of G the numbers assigned to the edges having c …
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