For a graph G=(V,E) and positive integer p, the exact distance-p graph
G[â™®p] is the graph with vertex set V and with an edge between
vertices x and y if and only if x and y have distance p. Recently,
there has been an effort to obtain bounds on the chromatic number
χ(G[♮p]) of exact distance-p graphs for G from certain
classes of graphs. In particular, if a graph G has tree-width t, it has
been shown that χ(G[♮p])∈O(pt−1) for odd p,
and χ(G[♮p])∈O(ptΔ(G)) for even p. We
show that if G is chordal and has tree-width t, then χ(G[♮p])∈O(pt2) for odd p, and χ(G[♮p])∈O(pt2Δ(G)) for even p.
If we could show that for every graph H of tree-width t there is a
chordal graph G of tree-width t which contains H as an isometric subgraph
(i.e., a distance preserving subgraph), then our results would extend to all
graphs of tree-width t. While we cannot do this, we show that for every graph
H of genus g there is a graph G which is a triangulation of genus g and
contains H as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which
arise from reviewers' comment