Some Existence Theorems on Path Factors with Given Properties in Graphs

A path factor of G is a spanning subgraph of G such that its each component is a path.A path factor is called a P≥n-factor if its each component admits at least n vertices.A graph G is called P≥n-factor covered if G admits a P≥n-factor containing e for any e∈E(G),which is defined by[Discrete Mathematics,309,2067-2076(2009)].We first define the concept of a(P≥n,k)-factor-critical covered graph,namely,a graph G is called(P≥n,k)-factor-critical covered if G-D is P≥n-factor covered for any D⊆V(G)with|D|=k.In this paper,we verify that(i)a graph G withκ(G)≥k+1 is(P≥2,k)-factor-critical covered if bind(G)>2+k/3;(ii)a graph G with|V(G)|≥k+3 andκ(G)≥k+1 is(P≥3,k)-factor-critical covered if bind(G)≥4+k/3.