The Binding Number of a Graph and Its Anderson Number*

The binding number of a graph G, bind(G), is defined; some examples of its calculation are given, and some upper bounds for it are proved. It is then proved that, if bind(G) > c, then G contains at least 1 G 1 c/(c + 1) disjoint edges if 0 < c < 3, at least ( G I (3c 2)/3c 2(c 1)/c disjoint edges if 1 < c < $, a Hamiltonian circuit if c > $, and a circuit of length at least 3(1 G / l)(c 1)/c if 1 < c < $ and G is not one of two specified exceptional graphs. Each of these results is best possible. The Anderson number of a graph is defined. The Anderson numbers of a few very simple graphs are determined; and some rather weak bounds are obtained, and some conjectures made, on the Anderson numbers of graphs in general.