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

Abstract

Let $G^w=(V,E,w)$ be a weighted graph, where $G=(V,E)$ is its underlying graph and $w:E\to [1,\infty )$ 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 $e=\mathit{uv}$ satisfies $\mathrm{dist}(c(u),c(v))⩾w(\mathit{uv})$. The smallest p allowing a p-coloring of $G^w$ 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.