New Diagonal Graph Ramsey Numbers of Unicyclic Graphs

Abstract

Grossman conjectured that $R(G, G) = 2 \cdot |V(G)| - 1$, for all simple connected unicyclic graphs $G$ of odd girth and $|V(G)| \geq 4$. In this note, we prove his conjecture for various classes of $G$ containing a triangle. In addition, new diagonal graph Ramsey numbers are calculated for some classes of simple connected unicyclic graphs of even girth.