Chromatically unique multibridge graphs (Q1422157)

Summary: Let \(\theta (a_1,a_2,\dots ,a_k)\) denote the graph obtained by connecting two distinct vertices with \(k\) independent paths of lengths \(a_1,a_2, \dots ,a_k\) respectively. Assume that \(2\leq a_1\leq a_2\leq \cdots \leq a_k\). We prove that the graph \(\theta (a_1,a_2, \dots ,a_k)\) is chromatically unique if \(a_k < a_1+a_2\), and find examples showing that \(\theta (a_1,a_2, \dots ,a_k)\) may not be chromatically unique if \(a_k=a_1+a_2\).