Structural properties of edge-chromatic critical multigraphs

Appearing in different format, Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) conjectured that if $G$ is an edge-$k$-critical graph with $k \ge \Delta +1$, then $|V(G)|$ is odd and, for every edge $e$, $E(G-e)$ is a union of disjoint near-perfect matchings, where $\Delta$ denotes the maximum degree of $G$. Tashkinov tree method shows that critical graphs contain a subgraph with two important properties named closed and elementary. Recently, efforts have been made in extending graphs beyond Tashkinov trees. However, these results can only keep one of the two essential properties. In this paper, we developed techniques to extend Tashkinov trees to larger subgraphs with both properties. Applying our result, we have improved almost all known results towards Goldberg's conjecture. In particular, we showed that Goldberg's conjecture holds for graph $G$ with $|V(G)| \le 39$ and $|\Delta(G)| \le 39$ and Jacobsen's equivalent conjecture holds for $m \le 39$ while the previous known bound is $23$.