Expanding graphs, Ramanujan graphs, and 1-factor perturbations

We construct (k+-1)-regular graphs which provide sequences of expanders by adding or substracting appropriate 1-factors from given sequences of k-regular graphs. We compute numerical examples in a few cases for which the given sequences are from the work of Lubotzky, Phillips, and Sarnak (with k-1 the order of a finite field). If k+1 = 7, our construction results in a sequence of 7-regular expanders with all spectral gaps at least about 1.52

On the Existence of General Factors in Regular Graphs

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.Comment: 10 page