Hadwiger's conjecture for circular colorings of edge-weighted graphs

Let G(w) = (V, E, w) be a weighted graph, where G = (V, E) is its underlying graph and w : E -> [1, infinity) is the edge weight function. A (circular) p-coloring of G(w) is a mapping c of its vertices into a circle of perimeter p so that every edge e = uv satisfies dist(c(u), c(v)) >= w(uv). The smallest p allowing ap-coloring of Gw is its circular chromatic number, chi(c)(G(w)). A p-basic graph is a weighted complete graph, whose edge weights satisfy triangular inequalities, and whose optimal traveling salesman tour has length p. Weighted Hadwiger's conjecture (WHC) at p >= 1 states that if p is the largest real number so that G(w) contains some p-basic graph as a weighted minor, then chi(c)(G(w)) <= p. We prove that WHC is true for p < 4 and false for p >= 4, and also that WHC is true for series-parallel graphs. (c) 2006 Elsevier B.V. All rights reserved.