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Lower Bounds and properties for the average number of colors in the non-equivalent colorings of a graph

Alain Hertz, Hadrien Mélot, Sébastien Bonte, Gauvain Devillez

TL;DR

This work conjecture several lower bounds on A(G) and prove that these conjectures are true for specific classes of graphs such as triangulated graphs and graphs with maximum degree at most 2.

Abstract

We study the average number $\mathcal{A}(G)$ of colors in the non-equivalent colorings of a graph $G$. We show some general properties of this graph invariant and determine its value for some classes of graphs. We then conjecture several lower bounds on $\mathcal{A}(G)$ and prove that these conjectures are true for specific classes of graphs such as triangulated graphs and graphs with maximum degree at most 2.

Lower Bounds and properties for the average number of colors in the non-equivalent colorings of a graph

TL;DR

This work conjecture several lower bounds on A(G) and prove that these conjectures are true for specific classes of graphs such as triangulated graphs and graphs with maximum degree at most 2.

Abstract

We study the average number of colors in the non-equivalent colorings of a graph . We show some general properties of this graph invariant and determine its value for some classes of graphs. We then conjecture several lower bounds on and prove that these conjectures are true for specific classes of graphs such as triangulated graphs and graphs with maximum degree at most 2.

Paper Structure

This paper contains 8 sections, 24 theorems, 52 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Basic properties of $\mathcal{A}(G)$
  4. Some values for $\mathcal{A}(G)$
  5. Lower bounds on $\mathcal{A}(G)$
  6. Conjectures
  7. Proof of the conjectures for graphs $G$ with $\Delta(G)\le 2$
  8. Concluding remarks

Key Result

Theorem 1

Given any two graphs $G_1$ and $G_2$, we have

Figures (2)

  • Figure 1: The non-equivalent colorings of $\overline{{\sf P}}_5$.
  • Figure 2: The three graphs that define the lower bounds $L_1(7)$, $L_2(7,4)$ and $L_3(7,4)$.

Theorems & Definitions (49)

  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5
  • proof
  • ...and 39 more