The concept of the binding number of a graph was introduced by Woodall in 1973. In this paper w e characterize the set F, of all pairs (a, b ) of integers such that there is a graph G with n vertices and binding number alb that has a realizing set of b vertices.
BINDING NUMBERS OF CERTAIN SPECIAL GRAPHSDefinition. (Woodall [2]) Let V = V(G) denote the set of vertices and E = E(G) the set of edges of the graph G. For x E V(G) let T ( x ) denote the set of all vertices of G which are adjacent to x. For X C V(G) let T(X) = UxaT(X). Let S(G) = {XI fl Z X C V(G), r(X) # V(G)). Then the binding number, bind(G), of G is bind(G) = min I T ( X ) I / I XI . X € 6 ( G )We introduce the term "realizing set" for a set X E S(G) such that bind(G)= I r(x>l /IXl.
Abstract. In this note, we derive a lower bound for the number of vertices of a graph in terms of its diameter, d, connectivity zc and minimum degree p which is sharper than that of Watkins [1] by an amount 2(p -k).Let G be any finite, undirected graph with neither loops nor multiple edges. Let zz, p, k and d denote the number of vertices, minimum degree, connectivity and diameter of G respectively. Watkins [1] has proved that if k > 1, then zz > k(d -1) + 2. He has used Menger's theorem to obtain the above result. In this note we prove a theorem from which Watkins' result follows as a corollary. Our proof is simple and elementary. Moreover the lower bound we obtain is sharper than that of Watkins by the amount 2(p -k).
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