Independence number and minimum degree for path-factor critical uniform graphs.

A P ≥ k -factor is a spanning subgraph H of G whose components are paths of order at least k. A graph G is P ≥ k -factor uniform if for arbitrary e 1 , e 2 ∈ E (G) with e 1 ≠ e 2 , G has a P ≥ k -factor containing e 1 and avoiding e 2. Liu first put forward the concept of (P ≥ k , n) -critical uniform graph, that is, a graph G is called (P ≥ k , n) -critical uniform if the graph G − V ′ is P ≥ k -factor uniform for any V ′ ⊆ V (G) with | V ′ | = n. In this paper, two new results on (P ≥ k , n) -critical uniform graphs (k = 2 , 3) in terms of independence number and minimum degree are presented. Furthermore, we show the sharpness of the main results in this paper by structuring special counterexamples. [ABSTRACT FROM AUTHOR]