- 著者
-
松田 晴英
- 出版者
- 中村学園大学
- 雑誌
- 中村学園大学・中村学園大学短期大学部研究紀要 (ISSN:13477331)
- 巻号頁・発行日
- vol.33, pp.221-224, 2001-03-15
Let n be an odd integer. A graph G is said to be m-factor-critical if G-H has a [1, n]-odd factor for each H⊂V(G) with |H|=m. In terms of neighborhood unions, we give a sufficient condition for a graph to be m-factor-critical with respect to [1, n]-odd factor. Let G be a k-connected graph. Let m be an integer with 0 ≤ m ≤ k and |G| ≡ m (mod 2), and let α be a real number with 1/(n+1) ≤ α ≤ 1. If |N_G(A)| > α(|G|-(n+1)k+nm-2)+k for every independent vertex set A of order [α(n(k-m)+2)], then G is m-factor-critical with respect to [1, n]-odd factor. We also discuss the sharpness of the result. x ∈ V(G), we denote by deg_G(x) the degree of x in G, and by N_G(x) the set of vertices adjacent to x in G. For a subset X ⊆ V(G), let N_G(X) = U_<x∈X>N_G(x). The number of odd components of odd in G is denoted by o(G). Let n be an odd integer. Then a spanning subgraph F of G is called a[1, n]-odd factor if deg_F(x) ∈{1, 3 ..., n} for all x ∈V(G). For a nonnegative integer m, a graph G is said to be m-factor-critical with respect to [1, n]-odd factor if G-H has a [1, n]-odd factor for each H ⊂ V(G) with |H| = m. Note that when n=1, [1, n]-odd factor nothing but 1-factor or perfect matching. Kano and Matsuda [3] introduce [1, n]-odd factor criticality, that is, it considers conditions for a proper subset of a graph to have a [1, n]-odd factor. One of results in the paper [3] is the following :