Graph Coloring · Superchallenge problem

Let $G$ have vertices $v_1,\ldots,v_5$ forming the $5$-cycle $v_1v_2,v_2v_3,v_3v_4,v_4v_5,v_5v_1$; vertices $u_1,\ldots,u_5$ where $u_i$ is adjacent to $v_{i-1}$ and $v_{i+1}$ (indices mod $5$) – the same two vertices $v_i$'s OWN cycle-neighbors are; and one more vertex $w$ adjacent to all five $u_i$ ($20$ edges, $11$ vertices in all). This graph has no triangle. What is $\chi(G)$?