Sufficient conditions for graphs to have strong parity factors

Abstract

A graph G has a strong parity factor F if for every subset X ⊆ V (G) with |X| even, G contains a spanning subgraph F such that δ(F) ≥ 1, dF (u) ≡ 1 (mod 2) for any u ∈ X, and dF(v) ≡ 0 (mod 2) for any v ∈ V(G) \ X. In this article, we first provide a size condition for a graph having a strong parity factor. Then we put forward a toughness condition to guarantee that a graph has a strong parity factor.