On the maximal number of disjoint circuits of a graph

Throughout this paper Gg" will denote a graph with n vertices and k edges where circuits consisting of two edges and loops (i . e. circuits of one edge) are not permitted and G'" will denote a graph of n vertices and k edges where loops and circuits with two edges are permitted . v(G) (respectively v(G)) will denote the number of edges of G (respectively G) . If x,, x" . . ., x,, are some of the vertices of G, then (G-x, . . . -xk) will denote the graph which we obtain from G by omitting the vertices x,, . . ., x k and all the edges incident to them . By G(x,, . . ., x k ) we denote the subgraph of G spanned by the vertices x,, . . ., xk . The valency of a vertex x v (x) will denote the number of edges incident to it. (A loop is counted doubly.) The edge connecting x, and x, will be denoted by [x,, x,], edges will sometimes be denoted by e,, ez , . . . . (x,, x,, . . .xk ) will denote the circuit having the edges [x,, x,], . . ., [xk_,, .vk], [x k x,] . A set of edges is called independent if no two of them have a common vertex . A set of circuits will be called independent if no two of them have a common vertex . They will be called weakly independent if no two of them have a common edge . In a previous paper ERDŐS and GALLAI [l] proved that every