Ramsey theory and strength of graphs (2408.01475v2)

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.