On the chromatic equivalence class of a family of graphs

Let Pn∗ denote the graph obtained by joining a new vertex to every vertex of a path on n vertices. Let Ui,j (n) denote the set of all connected graphs obtained from Pi∗UPj∗ by connecting the four vertices of degree 2 by two paths of lengths s(⩾0) and t(⩾1) such that s + t = n − i − j is a constant. Li and Whitehead Jr. conjecture that U3,4(n) forms a chromatic equivalence class by itself. In this note we prove the conjecture in the affirmative.