Tight toughness bounds for path-factor critical avoidable graphs

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+24; (ii) An (n+2)-connected graph is (P≥3,n)-factor critical avoidable if its isolated toughness I(G)>n+64.