On the Multigraph Overfull Conjecture

A subgraph $H$ of a multigraph $G$ is overfull if $|E(H) | > \Delta(G) \lfloor |V(H)|/2 \rfloor$. Analogous to the Overfull Conjecture proposed by Chetwynd and Hilton in 1986, Stiebitz et al. in 2012 formed the multigraph version of the conjecture as follows: Let $G$ be a multigraph with maximum multiplicity $r$ and maximum degree $\Delta>\frac{1}{3} r|V(G)|$. Then $G$ has chromatic index $\Delta(G)$ 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 $n$. (1) If $G$ is $k$-regular with $k\ge r(n/2+18)$, 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 $\delta(G)\ge r(n/2+18)$, then $\chi'(G)=\lceil \chi'_f(G) \rceil$, where $\chi'_f(G)$ is the fractional chromatic index of $G$. (3) If the minimum degree of $G$ is at least $(1+\varepsilon)rn/2$ for any $0<\varepsilon<1$ and $G$ contains no overfull subgraph, then $\chi'(G)=\Delta(G)$. The proof is based on the decomposition of multigraphs into simple graphs and we prove a slightly weak version of a conjecture due to the first author and Tipnis from 1991 on decomposing a multigraph into constrained simple graphs. The result is of independent interests.