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

Let Gw=(V,E,w)Gw=(V,E,w) be a weighted graph, where G=(V,E)G=(V,E) is its underlying graph and w:E→[1,∞)w:E→[1,∞) is the edge weight function. A (circular) p  -coloring of GwGw is a mapping c of its vertices into a circle of perimeter p   so that every edge e=uve=uv satisfies dist(c(u),c(v))⩾w(uv)dist(c(u),c(v))⩾w(uv). The smallest p allowing a p  -coloring of GwGw is its circular chromatic number, χc(Gw)χc(Gw).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⩾1p⩾1 states that if p   is the largest real number so that GwGw contains some p  -basic graph as a weighted minor, then χc(Gw)⩽pχc(Gw)⩽p.We prove that WHC is true for p<4p<4 and false for p⩾4p⩾4, and also that WHC is true for series–parallel graphs.