Hamilton-connected cores and five cycle--wheel Ramsey numbers

Abstract: Let $W_s=K_1+C_{s-1}$ denote the wheel on $s$ vertices. We give structural proofs that $R(C_{14},W_{11})=27$ and $R(C_{15},W_{11})=29$. Together with the theorem of Chen et al. for $n\ge16$, these equalities give $R(C_n,W_{11})=2n-1$ for every $n\ge14$. The two boundary values were included in an earlier survey announcement. We also give structural proofs of $R(C_8,W_7)=15$, $R(C_9,W_7)=17$, and $R(C_8,W_9)=15$. The common starting point is a Hamilton-connected core lemma. For the eleven-vertex wheel, bounds on vertex connectivity and on the matching number of a bipartite graph associated with a local cycle yield a vertex cut of order nine. Paths with prescribed endpoints then rule out every possible pair of orders of the two remaining vertex sets. For the smaller wheels, we use the structure of critical cycle colorings and local cycle-shortening arguments. We also give complete structural classifications of the $(C_8,C_6)$- and $(C_9,C_6)$-critical colorings, recovering the previously reported counts 24 and 26. All proofs are combinatorial and use no exhaustive graph enumeration.