Spanning subgraphs with specified valencies

The author has published a necessary and sufficient condition for a fmite loopless graph to have a spanning subgraph with a specified positive valency at each vertex (see [8,9). In the present paper it is contended that the condition can be made more useful as a tool of graph theory by imposing a maximality condition. 1. The condition for an I-factor Let G be a finite graph. Loops and multiple joins are allowed. Let I be a function from the vertex-set V(G) of G to the set of non-negative integers. We define an/-Iactor of G as a spanning subgraph F of G such that the valency of x in F is/(x) for each vertex x of G. We recall that the "valency" of a vertex x in a graph is the number of incident edges, loops being counted twice. Let us define a G-triple as an ordered triple (S, T, U), where S, T and U are disjoint subsets of V(G) whose union is V(G). Let x be a vertex of G, and Ya subset of V(G). If x is in Y, we define X(Y, x) as the number of links joining x to vertices in Y \ {x}, plus twice the number of loops incident with x. But if x is not in Y, we define X(Y, x) as the number of links joining x to vertices in Y. Let Y be any subset of V(G). Consider the subgraph of G induced by Y, that is, consisting of the vertices of Y, the loops on these vertices and the links with both ends in' Y. We refer to the components of this sub-graph simply as the "components of Y". Let B = (S, T, U) be a G-triple. We describe a component C of U as odd or even (with respect to B) according as the number