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

A signed graph is a graph G together with a signature σ : E (G) → { 1 , − 1 }. For a real number r ≥ 1 , C r is a circle of circumference r. For two points a , b ∈ 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 ∈ 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) → C r such that for each positive edge e = u v , d (mod r) (f (u) , f (v)) ≥ 1 , and for each negative edge e = u v , 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. [ABSTRACT FROM AUTHOR]