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Tight Bounds on the Chromatic Edge Stability Index of Graphs

Saieed Akbari, John Haslegrave, Mehrbod Javadi, Nasim Nahvi, Helia Niaparast

Abstract

The chromatic edge stability index $\mathrm{es}_{χ'}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a graph with smaller chromatic index. We give best-possible upper bounds on $\mathrm{es}_{χ'}(G)$ in terms of the number of vertices of degree $Δ(G)$ (if $G$ is Class 2), and the numbers of vertices of degree $Δ(G)$ and ${Δ(G)-1}$ (if $G$ is Class 1). If $G$ is bipartite we give an exact expression for $\mathrm{es}_{χ'}(G)$ involving the maximum size of a matching in the subgraph induced by vertices of degree $Δ(G)$. Finally, we consider whether a minimum mitigating set, that is a set of size $\mathrm{es}_{χ'}(G)$ whose removal reduces the chromatic index, has the property that every edge meets a vertex of degree at least $Δ(G)-1$; we prove that this is true for some minimum mitigating set of $G$, but not necessarily for every minimum mitigating set of $G$.

Tight Bounds on the Chromatic Edge Stability Index of Graphs

Abstract

The chromatic edge stability index of a graph is the minimum number of edges whose removal results in a graph with smaller chromatic index. We give best-possible upper bounds on in terms of the number of vertices of degree (if is Class 2), and the numbers of vertices of degree and (if is Class 1). If is bipartite we give an exact expression for involving the maximum size of a matching in the subgraph induced by vertices of degree . Finally, we consider whether a minimum mitigating set, that is a set of size whose removal reduces the chromatic index, has the property that every edge meets a vertex of degree at least ; we prove that this is true for some minimum mitigating set of , but not necessarily for every minimum mitigating set of .
Paper Structure (3 sections, 8 theorems, 4 equations, 4 figures)

This paper contains 3 sections, 8 theorems, 4 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Structure of minimum mitigating sets

Key Result

Theorem 1

V For a graph $G$, ${\Delta(G)\leq\chi^\prime(G)\leq\Delta(G)+1}$.

Figures (4)

  • Figure 1: A $3$-edge-colorable graph obtained by removing two edges from $P$.
  • Figure 2: The graph $Q\cong P\setminus\{e\}$.
  • Figure 3: A graph $G$ with $t_\Delta=1$ and $\mathrm{es}_{\chi'}(G)$ arbitrarily large.
  • Figure 4: The graphs $G'$ (solid edges) and one choice of $G$ (solid and dashed edges), with a $4$-edge-coloring of $G$.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Remark 1
  • Remark 2
  • Theorem 5
  • proof
  • Remark 3
  • ...and 7 more