On the Existence of General Factors in Regular Graphs

TL;DR

This paper investigates the conditions under which certain degree-constrained spanning subgraphs, called H-factors, exist in regular graphs, providing both counterexamples and sharp criteria that extend classical theorems.

Contribution

It constructs an r-regular graph lacking a specific H*-factor and establishes a precise condition for the existence of general H-factors in r, r+1-graphs, extending classical graph factor theorems.

Findings

Constructed an r-regular graph without some H*-factor.

Provided a sharp condition for H-factor existence in r, r+1-graphs.

Extended Thomassen's and Tutte's theorems to broader classes of graphs.

Abstract

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 Lov\'asz'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 $H$. The 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.