BINDING NUMBER FOR PATH-FACTOR UNIFORM GRAPHS

A path-factor of a graph G is a spanning subgraph of G whose components are paths. A P->= d-factor of a graph G is a path-factor of G whose components are paths with at least d vertices, where d >= 2 is an integer. A graph G is called a P->= d-factor uniform graph if for any two different edges e(1 )and e(2) of G, G admits a P->= d-factor containing e(1) and avoiding e(2). The binding number of G is defined by bind(G) = min{vertical bar N-G(X)vertical bar/vertical bar X vertical bar : empty set not equal X subset of V(G), N-G(X) not equal V(G)}. In this paper, we prove that (i) a 3-connected graph G is a P->= 2 -factor uniform graph if bind (G) > 1; (ii) a 3-connected graph G is a P->= 3-factor uniform graph if bind(G) > 10/7.