Graph Coloring · Medium problem

A central router $c$ connects directly to three devices $x,y,z$, and device $x$ also connects to one more device $w$ (a small tree network, $4$ links in all). Coloring in the order $w,y,z,x,c$ — each device getting the smallest color not already used by an earlier-colored neighbor — uses some number of colors. By how many more colors does this order use compared to the TRUE chromatic number of the network?