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The edge chromatic transformation index of graphs

Armen S. Asratian, Carl Johan Casselgren

TL;DR

This paper introduces $χ'_{trans}(G)$, the minimum number of color classes involved in transforming any two proper $χ'(G)$-edge colorings of a graph, via a sequence of intermediate colorings. It develops a constructive, combinatorial framework using $t$-alternating ears and correction steps to prove tight upper bounds for broad graph families (e.g., $χ'_{trans}(G)≤4$ for many block types; planar graphs $≤8$; Halin graphs $≤5$; and planar regular bipartite multigraphs with $χ'_{trans}=2$). It also identifies sharpness examples and discusses the vertex-coloring analogue, which exhibits a negative result, underscoring fundamental differences between edge- and vertex-color transformations. The results advance understanding of how far edge-coloring transformations can be constrained within specific graph classes and outline directions for resolving the existence of a universal transformation constant.

Abstract

Given a graph or multigraph $G$, let $χ'_{trans}(G)$ denote the minimum integer $n$ such that any proper $χ'(G)$--edge coloring of $G$ can be transformed into any other proper $χ'(G)$--edge coloring of $G$ by a series of transformations such that each of the intermediate colorings is a proper $χ'(G)$--edge coloring of $G$ and each of the transformations involves at most $n$ color classes of the previous coloring. We call $χ'_{trans}(G)$ 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 $χ'_{trans}(G)\leq 4$. This bound is sharp for series-parallel and wheel graphs. We also show that $χ'_{trans}(G)\leq 8$ for all planar graphs $G$, $χ'_{trans}(G)\leq 5$ if $G$ is a Halin graph and $χ'_{trans}(G)=2$ 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.

The edge chromatic transformation index of graphs

TL;DR

This paper introduces , the minimum number of color classes involved in transforming any two proper -edge colorings of a graph, via a sequence of intermediate colorings. It develops a constructive, combinatorial framework using -alternating ears and correction steps to prove tight upper bounds for broad graph families (e.g., for many block types; planar graphs ; Halin graphs ; and planar regular bipartite multigraphs with ). It also identifies sharpness examples and discusses the vertex-coloring analogue, which exhibits a negative result, underscoring fundamental differences between edge- and vertex-color transformations. The results advance understanding of how far edge-coloring transformations can be constrained within specific graph classes and outline directions for resolving the existence of a universal transformation constant.

Abstract

Given a graph or multigraph , let denote the minimum integer such that any proper --edge coloring of can be transformed into any other proper --edge coloring of by a series of transformations such that each of the intermediate colorings is a proper --edge coloring of and each of the transformations involves at most color classes of the previous coloring. We call the {\it edge chromatic transformation index of }. In this paper we show that if is a graph with maximum degree at least , 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 , then . This bound is sharp for series-parallel and wheel graphs. We also show that for all planar graphs , if is a Halin graph and if is a regular bipartite planar multigraph. Finally, we consider the analogous problem for vertex colorings, and show that for any there is an infinite class of graphs with chromatic number such that for every any two proper -vertex colorings of can be transformed to each other only by a transformation, involving all color classes.

Paper Structure

This paper contains 5 sections, 25 theorems, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and preliminary results
  3. Main lemmas
  4. Main results and proofs
  5. Concluding remarks

Key Result

Theorem 1.5

Let $q$ be an integer, $q\geq 3$, and $G$ be a Class 1 graph with $\Delta(G)\geq 5$. If every block of $G$ is either a bipartite graph, or a $(q+1)$-degenerate graphFor a positive integer $k$, a graph $G$ is said to be $k$-degenerate if for each subgraph $H$ of $G$, $H$ contains a vertex $x$ with $d

Figures (3)

  • Figure 1: A Halin graph $H$ with $\chi'_{trans}(H)=4$.
  • Figure 2: A series-parallel and outerplanar graph $G$ with $\chi'_{trans}(G)=4$.
  • Figure 3: A planar bipartite graph $G$ with $\chi'_{trans}(G)=3.$

Theorems & Definitions (48)

  • Definition
  • Theorem 1.5
  • Proposition 2.1: Fournier Fournier
  • Proposition 2.2
  • Proposition 2.3: Dirac dirac, Plummer plummer
  • Proposition 2.4: Machado et al machado
  • Proposition 2.5
  • Proposition 2.6: Bonduelle, Kardos̆ girth
  • Proposition 2.7
  • proof
  • ...and 38 more