Let $G$ be a graph and $H\colon V(G)\to 2^\mathbb{N}$ a set function associated with $G$. A spanning subgraph $F$ of $G$ is called an $H$-factor if the degree of any vertex $v$ in $F$ belongs to the set $H(v)$. This paper contains two results on the existence of $H$-factors in regular graphs. First, we construct an $r$-regular graph without some given $H^*$-factor. In particular, this gives a negative answer to a problem recently posed by Akbari and Kano. Second, by using Lovász's characterization theorem on the existence of $(g, f)$-factors, we find a sharp condition for the existence of general $H$-factors in $\{r, r+1\}$-graphs in terms of the maximum and minimum of $H$. This result reduces to Thomassen's theorem for the case that $H(v)$ consists of the same two consecutive integers for all vertices $v$ and to Tutte's theorem if the graph is regular in addition.

A (k, g)-cage is a k-regular graph with girth g that has the fewest number of vertices. It has been conjectured (Fu et al., J Graph Theory 24 (1997), 187–191) that all (k, g)-cages are k-connected for k ≥ 3. A connected graph G is said to be superconnected if every minimum cut-set S is the neighborhood of a vertex of minimum degree. Moreover, if G-S has precisely two components, then G is called tightly superconnected. It was shown (Xu et al., Ars Combin 64 (2002), 181–192) that every (4, g)-cage is 4-connected. In this article, we prove that every (4, g)-cage is tightly superconnected when g is even and g ≥ 12. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010