The edge chromatic transformation index of graphs

Given a graph or multigraph $G$, let $\chi'_{trans}(G)$ denote the minimum integer $n$ such that any proper $\chi'(G)$--edge coloring of $G$ can be transformed into any other proper $\chi'(G)$--edge coloring of $G$ by a series of transformations such that each of the intermediate colorings is a proper $\chi'(G)$--edge coloring of $G$ and each of the transformations involves at most $n$ color classes of the previous coloring. We call $\chi'_{trans}(G)$ the {\it edge chromatic transformation index of $G$}. In this paper we show that if $G$ is a graph with maximum degree at least $4$, where every block is either a bipartite graph, a series-parallel graph, a chordless graph, a wheel graph or a planar graph of girth at least $7$, then $\chi'_{trans}(G)\leq 4$. This bound is sharp for series-parallel and wheel graphs. We also show that $\chi'_{trans}(G)\leq 8$ for all planar graphs $G$, $\chi'_{trans}(G)\leq 5$ if $G$ is a Halin graph and $\chi'_{trans}(G)=2$ if $G$ is a regular bipartite planar multigraph. Finally, we consider the analogous problem for vertex colorings, and show that for any $k\geq 3$ there is an infinite class $\cal G$$(k)$ of graphs with chromatic number $k$ such that for every $G\in \cal G$$(k)$ any two proper $k$-vertex colorings of $G$ can be transformed to each other only by a transformation, involving all $k$ color classes.