On the Chromatic Roots of Generalized Theta Graphs

The generalized theta graph Θs1, …, 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(Θs1, …, 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 Θ2, …, 2≃K2, k indeed has a chromatic root of modulus [1+o(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.