Ramsey theory and strength of graphs

TL;DR

This paper explores the concept of graph strength, establishes bounds related to Ramsey numbers, and investigates the sum of strengths of a graph and its complement, providing new theoretical insights and open problems.

Contribution

It introduces new bounds and exact values for the strength of graphs and their complements, extending existing conditions and connecting to Ramsey theory.

Findings

Established a lower bound for related Ramsey numbers.

Proved a sharp lower bound for the function f(n).

Provided additional lower bounds and exact values for f(n).

Abstract

A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\left\{ 1,2,\ldots ,n\right\} $ to the vertices of $G$, where each $uv\in E\left( G\right) $ is labeled $f\left( u\right) +f\left( v\right) $. The strength $\mathrm{str}\left( G\right) $ of $G$ is defined by $\mathrm{str}\left( G\right) =\min \left\{ \mathrm{str}_{f}\left( G\right) \left\vert f\text{ is a numbering of }G\right. \right\}$, where $\mathrm{str}_{f}\left( G\right) =\max \left\{ f\left( u\right) +f\left( v\right) \left\vert uv\in E\left( G\right) \right. \right\} $. Let $f\left( n\right) $ denote the maximum of $\mathrm{str}\left( G\right) +% \mathrm{str}\left( \overline{G}\right) $ over nonempty graphs $G$ and $% \overline{G}$ of order $n$, where $\overline{G}$ represents the complement of $G$. In this paper, we establish a lower bound for the Ramsey numbers related to the concept of strength of a graph and show a sharp lower bound for $f\left( n\right) $. In addition to these results, we provide another lower bound and remark some exact values for $f\left( n\right) $. Furthermore, we extend existing necessary and sufficient conditions involving the strength of a graph. Finally, we investigate bounds for $\mathrm{str}\left( G\right) +\mathrm{str}\left( \overline{G}\right) $ whenever $G$ and $\overline{G}$ are nonempty graphs of order $n$. Throughout this paper, we propose some open problems arising from our study.