Sufficient conditions for the existence of pseudo 2-factors without isolated vertices and small odd cycles

A spanning subgraph F of a graph G is called a {P2,C2i+1:i≥k}-factor if each component of F is isomorphic to either a path of order 2 or a cycle of order 2i+1 for some i≥k. In this paper, we obtain the following two results (here ci(G−X) is the number of components C of G−X with |V(C)|=i):(i)If a graph G satisfies c1(G−X)+c3(G−X)≤12|X| for all X⊆V(G), then G has a {P2,C2i+1:i≥2}-factor.(ii)For k≥3, if a graph G satisfies ∑0≤j≤k−1c2j+1(G−X)≤25(k2−1)|X| for all X⊆V(G), then G has a {P2,C2i+1:i≥k}-factor.