Multicolor Ramsey Number of Caterpillar Graphs

We study multicolor Ramsey numbers and multicolor bipartite Ramsey numbers for caterpillar graphs, subdivided stars, and their combinations. A caterpillar graph $$C_p(a_1,\ldots,a_p)$$ is a tree formed by attaching $$a_i$$ leaves to each vertex $$i$$ of a central path $$P_p$$ . Building on results, such as the exact determination of multicolor Ramsey number of double stars, we extend these findings to more complex families, including $$C_3(n,m,t)$$ , caterpillar graphs with three central vertices. For sufficiently large $$n+t$$ relative to $$m$$ and odd $$k$$ , we establish tight bounds and exact values of $$R_k(C_3(n,m,t))$$ . We further study the graph $$C_2(S^{n_1}_n,m)$$ which contains a central edge, where one vertex supports a subdivided star $$S^{n_1}_n$$ and the other one supports a star $$K_{1,m}$$ . Under suitable conditions, we derive the exact value of the multicolor Ramsey number of $$C_2(S^{n_1}_n,m)$$ , improving known results for subdivided stars and their combinations. In bipartite settings, we determine the exact value of multicolor bipartite Ramsey number of both $$S^{n_1}_n$$ and $$C_2(S^{n_1}_n,m)$$ , providing results that extend and refine prior work on bipartite double stars.