Tight toughness bounds for path-factor critical avoidable graphs

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 P ≥ k -factor avoidable if for every edge e ∈ E ( G ) , G has a P ≥ k -factor excluding e. A graph G is said to be ( P ≥ k , n ) -factor critical avoidable if the graph G − V ′ is P ≥ k -factor avoidable for any V ′ ⊆ V ( G ) with | V ′ | = n . Here we study the sharp bounds of toughness and isolated toughness conditions for the existence of ( P ≥ 3 , n ) -factor critical avoidable graphs. In view of graph theory approaches, this paper mainly contributes to verify that (i) An ( n + 2 ) -connected graph is ( P ≥ 3 , n ) -factor critical avoidable if its toughness τ ( G ) > n + 2 4 ; (ii) An ( n + 2 ) -connected graph is ( P ≥ 3 , n ) -factor critical avoidable if its isolated toughness I ( G ) > n + 6 4 .