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Using Graph Theory to Derive Inequalities for the Bell Numbers

Alain Hertz, Anaelle Hertz, Hadrien Mélot

TL;DR

This work studies properties on the average number of colors in the non-equivalent colorings of a graph to discover new non trivial inequalities for the Bell numbers.

Abstract

The Bell numbers count the number of different ways to partition a set of $n$ elements while the graphical Bell numbers count the number of non-equivalent partitions of the vertex set of a graph into stable sets. This relation between graph theory and integer sequences has motivated us to study properties on the average number of colors in the non-equivalent colorings of a graph to discover new non trivial inequalities for the Bell numbers. Example are given to illustrate our approach.

Using Graph Theory to Derive Inequalities for the Bell Numbers

TL;DR

This work studies properties on the average number of colors in the non-equivalent colorings of a graph to discover new non trivial inequalities for the Bell numbers.

Abstract

The Bell numbers count the number of different ways to partition a set of elements while the graphical Bell numbers count the number of non-equivalent partitions of the vertex set of a graph into stable sets. This relation between graph theory and integer sequences has motivated us to study properties on the average number of colors in the non-equivalent colorings of a graph to discover new non trivial inequalities for the Bell numbers. Example are given to illustrate our approach.

Paper Structure

This paper contains 4 sections, 19 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Some values and properties of $\mathcal{A}(G)$
  3. Inequalities for the Bell numbers
  4. Conclusion

Key Result

Proposition 1

If $G$ has a dominating vertex $v$, then $\mathcal{A}(G)=1+\mathcal{A}(G-v)$.

Figures (2)

  • Figure 1: The non-equivalent colorings of $C_{5}$.
  • Figure 2: The graph $H_{n,r}$

Theorems & Definitions (37)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • ...and 27 more