On the Ramsey Numbers of Certain Graphs of Order Five versus All Connected Graphs of Order Six

The Ramsey numbers \( r(F, G) \) are investigated, where \( F \) is a non-tree graph of order \( 5 \) and minimum degree \( 1 \), and \( G \) is a connected graph of order \( 6 \). For all pairs \( (F, G) \) where \( F \neq K_5 – K_{1,3} \), the exact value of \( r(F, G) \) is determined. In order to settle \( F = K_5 – K_{1,3} \), we prove \( r(K_5 – K_{1,3}, G) = r(K_4, G) \).