Abstrakt
A signed graph is a graph G together with a signature σ : E(G) RIGHTWARDS ARROW {1,-1}. For a real number r >= 1, C^r is a circle of circumference r. For two points a,b ELEMENT OF C^r, the distance d _{(mod r)}(a,b) between a and b is the length of the shorter arc of C^r connecting a and b.
For x \in C^r, the antipodal -x of x is the unique point in C^r of distance r/2 from x. A circular r-colouring of (G,σ) is a mapping f : V(G) \to C^r such that for each positive edge e = uv, d _{(mod r)}(f (u),f (v)) >= 1, and for each negative edge e = uv, d_{ (mod r)}(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 SP_g 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 χ_c(G,σ) of signed graphs in SP_g.