The circular chromatic number of signed series–parallel graphs of given girth

Abstract

A signed graph is a graph G together with a signature σ:E(G)→{1,−1}. For a real number r≥1, Cr is a circle of circumference r. For two points a,b∈Cr, the distance d(modr)(a,b) between a and b is the length of the shorter arc of Cr connecting a and b. For x∈Cr, the antipodal x̄ of x is the unique point in Cr of distance r/2 from x. A circular r-colouring of (G,σ) is a mapping f:V(G)→Cr such that for each positive edge e=uv, d(modr)(f(u),f(v))≥1, and for each negative edge e=uv, d(modr)(f(u),f(v)¯)≥1. The circular chromatic number of a signed graph (G,σ) is the minimum r such that (G,σ) is circular r–colourable. Let g∗(G,σ) be the length of the shortest cycle in (G,σ) with an odd number of positive edges. For a positive integer g, let SPg be the family of signed series–parallel graphs with g∗(G,σ)≥g. This paper determines, for any positive integer g, the supremum value of χc(G,σ) of signed graphs in SPg.