Abstract
Given a graph G and an integer k ≥ 2. A spanning subgraph H of G is called a P≥k-factor of G if every component of H is a path with at least k vertices. A graph G is said to be P≥k-factor covered if for any e ∈ E(G), G admits a P≥k-factor including e. A graph G is called a (P≥k, n)-factor critical covered graph if G – V′ is P≥k-factor covered for any V′ ⊆ V(G) with |V′| = n. In this paper, we study the toughness and isolated toughness conditions for (P≥k, n)-factor critical covered graphs, where k = 2, 3. Let G be a (n + 1)-connected graph. It is shown that (i) G is a (P≥2, n)-factor critical covered graph if its toughness $ \tau (G)>\frac{n+2}{3}$; (ii) G is a (P≥2, n)-factor critical covered graph if its isolated toughness $ I(G)>\frac{n+1}{2}$; (iii) G is a (P≥3, n)-factor critical covered graph if $ \tau (G)>\frac{n+2}{3}$ and |V(G)| ≥ n + 3; (iv) G is a (P≥3, n)-factor critical covered graph if $ I(G)>\frac{n+3}{2}$ and |V(G)| ≥ n + 3. Furthermore, we claim that these conditions are best possible in some sense.
Funder
National Natural Science Foundation of China
Subject
Management Science and Operations Research,Computer Science Applications,Theoretical Computer Science