On the chromatic roots of generalized theta graphs

Abstract

The generalized theta graph χS1,⋯,Sk consists of a pair of endvertices joined by k internally disjoint paths of lengths S1,⋯,Sk≥1. We prove that the roots of the chromatic polynomial π(χS1,⋯,Sk, Z) of a k-ary generalized theta graph all lie in the disc z - 1 ≤ [ 1 + o (1) ]k/log k, uniformly in the path lengths Si. Moreover, we prove that χ2,⋯,2≃K2indeed has a chromatic root of modulus [1+0(1)]k/log k. Finally, for k ≤ 8 we prove that the generalized theta graph with a chromatic root that maximizes z - 1 is the one with all path lengths equal to 2; we conjecture that this holds for all k. © 2001 Academic Press.