On the Multigraph Overfull Conjecture

A subgraph $H$ of a multigraph $G$ is overfull if $\mid E\left(H\right)\mid >\Delta \left(G\right)\lfloor \mid V\left(H\right)\mid /2\rfloor .$ Analogous to the Overfull Conjecture proposed by Chetwynd and Hilton in 1986, Stiebitz et al. formed the multigraph version of the conjecture as follows: Let $G$ be a multigraph with maximum multiplicity $r$ and maximum degree $\Delta >\frac{1}{3}r\mid V\left(G\right)\mid$. Then $G$ has chromatic index $\Delta \left(G\right)$ if and only if $G$ contains no overfull subgraph. In this paper, we prove the following three results toward the Multigraph Overfull Conjecture for sufficiently large and even $n$, where $n=\mid V\left(G\right)\mid$. (1) If $G$ is $k$-regular with $k\ge r\left(n/2+18\right)$, then $G$ has a 1-factorization. This result also settles a conjecture of the first author and Tipnis from 2001 up to a constant error in the lower bound of $k$. (2) If $G$ contains an overfull subgraph and $\delta \left(G\right)\ge r\left(n/2+18\right)$, then ${\chi }^{\prime }\left(G\right)=\lceil {\chi }_f^{\prime }\left(G\right)\rceil$, where ${\chi }_f^{\prime }\left(G\right)$ is the fractional chromatic index of $G$. (3) If the minimum degree of $G$ is at least $\left(1+\epsilon \right)rn/2$ for any $0<\epsilon <1$ and $G$ contains no overfull subgraph, then ${\chi }^{\prime }\left(G\right)=\Delta \left(G\right)$. The proof is based on the decomposition of multigraphs into simple graphs and we prove a slightly weaker version of a conjecture due to the first author and Tipnis from 1991 on decomposing a multigraph into constrained simple graphs. The result is also of independent interest.