On the chromatic roots of generalized theta graphs

Abstract

The generalized theta graph Θ_{s_1,...,s_k} consists of a pair of endvertices joined by k internally disjoint paths of lengths s_1,...,s_k \ge 1. We prove that the roots of the chromatic polynomial $pi(Θ_{s_1,...,s_k},z) of a k-ary generalized theta graph all lie in the disc |z-1| \le [1 + o(1)] k/\log k, uniformly in the path lengths s_i. Moreover, we prove that Θ_{2,...,2} \simeq K_{2,k} indeed has a chromatic root of modulus [1 + o(1)] k/\log k. Finally, for k \le 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.