For a graph G and an integer k 2, a P ₊ -factor of G is a spanning subgraph of G with each component isomorphic to some path of order at least k. A graph G is P ₊ -factor uniform if for any distinct edges e1 and e2, G admits a P ₊ -factor including e₁ and excluding e₂. For a non-negative integer n, G is ( (P ₊, n) ) -critical uniform if for any V′ ⊆ V (G) with |V′| = n, G-V′ is P ₊ -factor uniform. G is (P ₊, n) -critical deleted if for any V′ ⊆ V (G) with |V′| = n and e ∈ E (G-V′), G-V′-e contains a P ₊ -factor. In this note, we give some binding number and degree conditions for a graph to be ( (P ₂, n) ) -critical uniform and ( (P ₃, n) ) -critical deleted, which improve some known results.
Ping Zhang (Wed,) studied this question.