Graphs with small bandwidth and cutwidth

We give counter-examples to the following conjecture which arose in the study of small bandwidth graphs. “For a graph G, suppose that IV(G’)( < 1 + ci . diameter (G’) for any connected subgraph G’ of G, and that G does not contain any refinement of the complete binary tree of cz levels. Is it true that the bandwidth of G can be bounded above by a constant c depending only on c, and c,?” On the other hand, we show that if the maximum degree of G is bounded and G does not contain any refinement of a complete binary tree of a specified size, then the cutwidth and the topological bandwidth of G are also bounded.