Kempe Equivalent List Edge-Colorings of Planar Graphs

For a list assignment $L$ and an $L$-coloring $\varphi$, a Kempe swap in $\varphi$ is \emph{$L$-valid} if it yields another $L$-coloring. Two $L$-colorings are \emph{$L$-equivalent} if we can form one from another by a sequence of $L$-valid Kempe swaps. And a graph $G$ is \emph{$L$-swappable} if every two of its $L$-colorings are $L$-equivalent. We consider $L$-swappability of line graphs of planar graphs with large maximum degree. Let $G$ be a planar graph with $\Delta(G)\ge 9$ and let $H$ be the line graph of $G$. If $L$ is a $(\Delta(G)+1)$-assignment to $H$, then $H$ is $L$-swappable. Let $G$ be a planar graph with $\Delta(G)\ge 15$ and let $H$ be the line graph of $G$. If $L$ is a $\Delta(G)$-assignment to $H$, then $H$ is $L$-swappable. The first result is analogous to one for $L$-choosability by Borodin, which was later strengthened by Bonamy. The second result is analogous to another for $L$-choosability by Borodin, which was later strengthened by Borodin, Kostochka, and Woodall.Comment: 15 pages, 5 figures, 2 page appendix; to appear in Discrete Math (special issue in honor of Landon Rabern

Solution of Vizing's Problem on Interchanges for Graphs with Maximum Degree 4 and Related Results

Let $G$ be a Class 1 graph with maximum degree $4$ and let $t\geq 5$ be an integer. We show that any proper $t$-edge coloring of $G$ can be transformed to any proper $4$-edge coloring of $G$ using only transformations on $2$-colored subgraphs (so-called interchanges). This settles the smallest previously unsolved case of a well-known problem of Vizing on interchanges, posed in 1965. Using our result we give an affirmative answer to a question of Mohar for two classes of graphs: we show that all proper $5$-edge colorings of a Class 1 graph with maximum degree 4 are Kempe equivalent, that is, can be transformed to each other by interchanges, and that all proper 7-edge colorings of a Class 2 graph with maximum degree 5 are Kempe equivalent