Star-Wheel Ramsey Numbers

Abstract

For given graphs \( G \) and \( H \), the Ramsey number \( R(G, H) \) is the smallest natural number \( n \) such that for every graph \( F \) of order \( n \): either \( F \) contains \( G \) or the complement of \( F \) contains \( H \).

This paper investigates the Ramsey number \( R(S_n, W_m) \) of stars versus wheels. We show that if \( m \) is odd, \( n \geq 3 \), and \( m \leq 2n-1 \), then \( R(S_n, W_m) = 3n-2 \). Furthermore, if \( n \) is odd and \( n \geq 5 \), then \( R(S_n, W_m) = 3n – \mu \), where \( \mu = 4 \) if \( m = 2n – 4 \) and \( \mu = 6 \) if \( m = 2n – 8 \) or \( m = 2n – 6 \).