An Extension of Cui-Kano's Characterization on Graph Factors

Let G be a graph with vertex set V(G) and let H:V(G)→2N be a set function associated with G. An H‐factor of graph G is a spanning subgraphs F such that dF(v)∈H(v)foreveryv∈V(G).Let f:V(G)→N be an even integer‐valued function such that f≥4 and let Hf(v)={1,3,...,f(v)−1,f(v)} for v∈V(G). In this article, we investigate Hf‐factors of graphs by using Lovász's structural descriptions. Let o(G) denote the number of odd components of G. We show that if one of the following conditions holds, then G contains an Hf‐factor. (i)|V(G)| is even and o(G−S)≤f(S) for all S⊆V(G); (ii)|V(G)| is odd, dG(v)≥f(v)−1 for all v∈V(G) and o(G−S)≤f(S) for all ∅≠S⊆V(G). As a corollary, we show that if a graph G of odd order with minimum degree at least 2n−1 satisfies o(G−S)≤2n|S|forall∅≠S⊆V(G),then G contains an Hn‐factor, where Hn={1,3,...,2n−1,2n}. In particular, we make progress on the characterization problem for a special family of graphs proposed by Akiyama and Kano.