On the Ramsey numbers for a combination of paths and Jahangirs

TL;DR

This paper advances the understanding of Ramsey numbers involving paths and Jahangir graphs, providing improved bounds and exact values for specific graph combinations, which deepens the theoretical knowledge in graph Ramsey theory.

Contribution

The paper improves previous bounds on the Ramsey number for paths versus Jahangir graphs and determines the exact Ramsey number for unions of paths and Jahangir graphs.

Findings

Improved the upper bounds on R(Path, Jahangir)

Determined the exact R(Union of Path, Jahangir)

Extended results to unions involving Jahangir graphs

Abstract

For given graphs $G$ and $H,$ the \emph{Ramsey number} $R(G,H)$ is the least natural number $n$ such that for every graph $F$ of order $n$ the following condition holds: either $F$ contains $G$ or the complement of $F$ contains $H.$ In this paper, we improve the Surahmat and Tomescu's result \cite{ST:06} on the Ramsey number of paths versus Jahangirs. We also determine the Ramsey number $R(\cup G,H)$, where $G$ is a path and $H$ is a Jahangir graph.