REMARKS ON PATH FACTORS IN GRAPHS

A spanning subgraph of a graph is defined as a path factor of the graph if its component are paths. A P->= n-factor means a path factor with each component having at least n vertices. A graph G is defined as a (P->= n,m)-factor deleted graph if G - E' has a P->= n-factor for every E' subset of E(G) with vertical bar E'vertical bar = m. A graph G is defined as a (P->= n, k)-factor critical graph if after deleting any k vertices of G the remaining graph of G admits a P->= n-factor. In this paper, we demonstrate that (i) a graph G is (P->= 3,m)-factor deleted if kappa(G) >= 2m + 1 and bind(G) >= 3/2 - 1/4m+4; (ii) a graph G is (P->= 3, k)-factor critical if kappa(G) >= k + 2 and bind(G) >= 5+k/4.