On the Existence of General Factors in Regular Graphs

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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.