On path-quasar Ramsey numbers

Abstract

Let \(G_1\) and \(G_2\) be two given graphs. The Ramsey number \(R(G_1,G_2)\) is the least integer \(r\) such that for every graph \(G\) on \(r\) vertices, either \(G\) contains a \(G_1\) or \(\overline{G}\) contains a \(G_2\). Parsons gave a recursive formula to determine the values of \(R(P_n,K_{1,m})\), where \(P_n\) is a path on \(n\) vertices and \(K_{1,m}\) is a star on \(m+1\) vertices. In this note, we study the Ramsey numbers \(R(P_n,K_1\vee F_m)\), where \(F_m\) is a linear forest on \(m\) vertices. We determine the exact values of \(R(P_n,K_1\vee F_m)\) for the cases \(m\leq n\) and \(m\geq 2n\), and for the case that \(F_m\) has no odd component. Moreover, we give a lower bound and an upper bound for the case \(n+1\leq m\leq 2n-1\) and \(F_m\) has at least one odd component.