A proof of a conjecture on matching-path connected size Ramsey number

Abstract

<abstract><p>For two graphs $ G_1 $ and $ G_2 $, the connected size Ramsey number $ {\hat{r}}_c(G_1, G_2) $ is the smallest number of edges of a connected graph $ G $ such that if each edge of $ G $ is colored red or blue, then $ G $ contains either a red copy of $ G_1 $ or a blue copy of $ G_2 $. Let $ nK_2 $ be a matching with $ n $ edges and $ P_4 $ a path with four vertices. Rahadjeng, Baskoro, and Assiyatun [Procedia Comput. Sci. 74 (2015), 32-37] conjectured that $ \hat{r}_{c}(nK_2, P_4) = 3n-1 $ if $ n $ is even, and $ \hat{r}_{c}(nK_2, P_4) = 3n $ otherwise. We verify the conjecture in this short paper.</p></abstract>