Hadwigerʹs conjecture for circular colorings of edge-weighted graphs Original Research Article

Let image be a weighted graph, where image is its underlying graph and image is the edge weight function. A (circular) p-coloring of image is a mapping c of its vertices into a circle of perimeter p so that every edge image satisfies image. The smallest p allowing a p-coloring of image is its circular chromatic number, image. 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 Hadwigerʹs conjecture (WHC) at image states that if p is the largest real number so that image contains some p-basic graph as a weighted minor, then image. We prove that WHC is true for image and false for image, and also that WHC is true for series–parallel graphs.