Conjectures on uniquely 3-edge-colorable graphs

A graph $G$ is {\it uniquely k-edge-colorable} if the chromatic index of $G$ is $k$ and every two $k$-edge-colorings of $G$ produce the same partition of $E(G)$ into $k$ independent subsets. For any $k\ne 3$, a uniquely $k$-edge-colorable graph $G$ is completely characterized; $G\cong K_2$ if $k=1$, $G$ is a path or an even cycle if $k=2$, and $G$ is a star $K_{1,k}$ if $k\geq 4$. On the other hand, there are infinitely many uniquely 3-edge-colorable graphs, and hence, there are many conjectures for the characterization of uniquely 3-edge-colorable graphs. In this paper, we introduce a new conjecture which connects conjectures of uniquely 3-edge-colorable planar graphs with those of uniquely 3-edge-colorable non-planar graphs.