The edge chromatic transformation index of graphs

TL;DR

This paper introduces the edge chromatic transformation index of graphs, providing bounds for various classes of graphs and exploring analogous vertex coloring transformations, advancing understanding of coloring reconfiguration.

Contribution

It defines the edge chromatic transformation index and establishes bounds for different graph classes, also analyzing vertex coloring transformations.

Findings

Bound of 4 for graphs with maximum degree ≥ 4 and specific block types

Bound of 8 for all planar graphs

Bound of 5 for Halin graphs and 2 for regular bipartite planar multigraphs

Abstract

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.