On the Ramsey numbers for paths and generalized Jahangir graphs

TL;DR

This paper determines the Ramsey numbers involving paths and generalized Jahangir graphs, providing exact values for these combinatorial parameters and extending previous results in graph theory.

Contribution

The paper establishes the exact Ramsey numbers for paths versus generalized Jahangir graphs and for multiple disjoint paths versus such graphs, advancing understanding in graph Ramsey theory.

Findings

Calculated $R(P_n, J_{s,m})$ for specific parameters.

Derived $R(tP_n, J_{s,m})$ for generalized Jahangir graphs.

Extended known bounds to exact values for these Ramsey numbers.

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 determine the Ramsey number of paths versus generalized Jahangir graphs. We also derive the Ramsey number $R(tP_n,H)$, where $H$ is a generalized Jahangir graph $J_{s,m}$ where $s\geq2$ is even, $m\geq3$ and $t\geq1$ is any integer.