Factorizations of some weighted spanning tree enumerators

We give factorizations for weighted spanning tree enumerators of Cartesian products of complete graphs, keeping track of fine weights related to degree sequences and edge directions. Our methods combine Kirchhoff's Matrix-Tree Theorem with the technique of identification of factors.Comment: Final version, 12 pages. To appear in the Journal of Combinatorial Theory, Series A. The paper has been reorganized, and the proof of Theorem 4 shortened, in light of a more general result appearing in reference [6

The Hamilton-Waterloo Problem with even cycle lengths

The Hamilton-Waterloo Problem HWP$(v;m,n;\alpha,\beta)$ asks for a 2-factorization of the complete graph $K_v$ or $K_v-I$, the complete graph with the edges of a 1-factor removed, into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors, where $3 \leq m < n$. In the case that $m$ and $n$ are both even, the problem has been solved except possibly when $1 \in \{\alpha,\beta\}$ or when $\alpha$ and $\beta$ are both odd, in which case necessarily $v \equiv 2 \pmod{4}$. In this paper, we develop a new construction that creates factorizations with larger cycles from existing factorizations under certain conditions. This construction enables us to show that there is a solution to HWP$(v;2m,2n;\alpha,\beta)$ for odd $\alpha$ and $\beta$ whenever the obvious necessary conditions hold, except possibly if $\beta=1$; $\beta=3$ and $\gcd(m,n)=1$; $\alpha=1$; or $v=2mn/\gcd(m,n)$. This result almost completely settles the existence problem for even cycles, other than the possible exceptions noted above

Vertex-regular 11-factorizations in infinite graphs

The existence of $1$-factorizations of an infinite complete equipartite graph $K_m[n]$ (with $m$ parts of size $n$) admitting a vertex-regular automorphism group $G$ is known only when $n=1$ and $m$ is countable (that is, for countable complete graphs) and, in addition, $G$ is a finitely generated abelian group $G$ of order $m$. In this paper, we show that a vertex-regular $1$-factorization of $K_m[n]$ under the group $G$ exists if and only if $G$ has a subgroup $H$ of order $n$ whose index in $G$ is $m$. Furthermore, we provide a sufficient condition for an infinite Cayley graph to have a regular $1$-factorization. Finally, we construct 1-factorizations that contain a given subfactorization, both having a vertex-regular automorphism group