# Using Graph Theory to Derive Inequalities for the Bell Numbers

@article{Hertz2021UsingGT, title={Using Graph Theory to Derive Inequalities for the Bell Numbers}, author={Alain Hertz and Anaelle Hertz and Hadrien M'elot}, journal={ArXiv}, year={2021}, volume={abs/2104.00552} }

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.

## 2 Citations

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

- Mathematics, Computer ScienceArXiv
- 2021

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.

Upper bounds on the average number of colors in the non-equivalent colorings of a graph

- Computer Science, MathematicsArXiv
- 2021

A general upper bound on A(G), the average number of colors in the non-equivalent colorings of a graph G, is given that is valid for all graphs G and a more precise one for graphs G of order n and maximum degree.

## References

SHOWING 1-10 OF 12 REFERENCES

On the Average Rank of an Element in a Filter of the Partition Lattice

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. A
- 1994

It is proved that (1 |F| ) Σ πϵF r(π) ≧ ( 1 |P n | ) Φ p(A, λ + 1)/(λ + 1) n holds for all positive integers λ.

On the Number of Distinct Block Sizes in Partitions of a Set

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. A
- 1985

The average number of distinct block sizes in a partition of a set of n elements is asymptotic to e log n as n → ∞, which is in striking contrast to the fact that the average total number of blocks in a partitions is ∼ n (log n ) −1 as n→ ∞.

A sharp lower bound on the number of non-equivalent colorings of graphs of order and maximum degree

- Mathematics, Computer ScienceDiscret. Appl. Math.
- 2018

A sharp lower bound on B ( G ) is determined for graphs G of order n and maximum degree n − 3 , and the graphs for which the bound is attained are characterized.

Counting the number of non-equivalent vertex colorings of a graph

- Computer Science, MathematicsDiscret. Appl. Math.
- 2016

Borders on B ( G ) for graphs with a maximum degree constraint are studied and all graphs that reach the bound with equality are described.

Stirling numbers of the second kind and Bell numbers for graphs

- Computer ScienceAustralas. J Comb.
- 2014

This paper summarizes the known properties of Stirling numbers of the second kind and Bell numbers for graphs, and proves new results about them, which give an alternative way to study r-StirlingNumbers of the first kind and r-Bell numbers.

Stirling Numbers of Forests and Cycles

- Mathematics, Computer ScienceElectron. J. Comb.
- 2013

This note considers Stirling numbers of forests, and establishes asymptotic normality for the number of classes in a uniformly chosen partition of $C_n$ (the cycle on $n$ vertices) into non-empty independent sets.

Bell and Stirling numbers for disjoint unions of graphs

- Congressus Numerantium
- 2010

Bell and Stirling Numbers for Graphs

- Mathematics
- 2009

The Bell number B(G) of a simple graph G is the number of partitions of its vertex set whose blocks are independent sets of G. The number of these partitions with k blocks is the (graphical) Stirling…

Chromatic polynomials and chro-maticity of graphs

- Mathematics
- 2005

# The Number of -Colourings and Its Enumerations # Chromatic Polynomials # Chromatic Equivalence of Graphs # Chromaticity of Multi-Partite Graphs # Chromaticity of Subdivisions of Graphs # Graphs in…

Engel's Inequality for Bell Numbers

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. A
- 1995

It is proved that Engel's conjecture for all n sufficiently large by an extension of the Moser-Wyman asymptotic formula for the Bell numbers is proved.