Strong Tutte Type Conditions and Factors of Graphs

Let odd(G) denote the number of odd components of a graph G and k ≥ 2 be an integer. We give sufficient conditions using odd(G − S) for a graph G to have an even factor. Moreover, we show that if a graph G satisfies odd(G − S) ≤ max{1, (1/k)|S|} for all S ⊂ V (G), then G has a (k − 1)-regular factor for k ≥ 3 or an H-factor for k = 2, where we say that G has an H-factor if for every labeling h : V (G) → {red, blue} with #{v ∈ V (G) : f(v) = red} even, G has a spanning subgraph F such that deg_F (x) = 1 if h(x) = red and deg_F (x) ∈ {0, 2} otherwise.