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 with k(G) ≥ k + 1 is (P⊆2, k)-factor-critical covered if bind $$\left( G \right) > {{2 + k} \over 3}$$ ; (ii) a graph G with ∣V(G)∣ ≥ k + 3 and k(G) ≥ k + 1 is (P≥3, k)-factor-critical covered if bind $$\left( G \right) \ge {{4 + k} \over 3}$$ .