Some existence theorems on path-factor critical avoidable graphs

A spanning subgraph F of G is called a path factor if every component of F is a path of order at least 2. Let k ≥ 2 be an integer. A P ≥ k -factor of G means a path factor in which every component has at least k vertices. A graph G is called a P ≥ k -factor avoidable graph if for any e ∈ E ( G ), G has a P ≥ k -factor avoiding e . A graph G is called a ( P ≥ k , n )-factor critical avoidable graph if for any W ⊆ V ( G ) with |W| = n , G − W is a P ≥ k -factor avoidable graph. In other words, G is ( P ≥ k , n )-factor critical avoidable if for any W ⊆ V ( G ) with |W| = n and any e ∈ E ( G − W ), G − W − e admits a P ≥ k -factor. In this article, we verify that (i) an ( n + r + 2)-connected graph G is ( P ≥2 , n )-factor critical avoidable if I ( G )>( n + r +2)/(2( r +2)) ; (ii) an ( n + r + 2)-connected graph G is ( P ≥3 , n )-factor critical avoidable if t ( G )>( n + r +2)/(2( r +2)) ; (iii) an ( n + r + 2)-connected graph G is ( P ≥3 , n )-factor critical avoidable if I ( G )>( n +3( r +2))/(2( r +2)) ; where n and r are two nonnegative integers.