I have a problem like this:
An undirected graph $G$ has width $w$ if the vertices can be arranged in a sequence $v_1, v_2, v_3, ..., v_n$ such that each vertex $v_i$ is joined by an edge to at most $w$ preceding vertices. (Vertex $v_j$ precedes $v_i$ if $j$ < $i$.) Use induction to prove that every graph with width at most $w$ is $(w + 1)$-colorable.
I think I could do the proof in general, but before that, I would need to understand what graph width is. My problem is, I can't really visualize how a graph that satisfies the property from above (the vertices can be arranged in the given sequence) would look like. I also read the definition of Treewidth, but it wasn't clear for me if that's even the same thing as in the problem.
How would a graph look like that satisfies the property from the problem description?