On the chromaticity of certain subgraphs of a q‐tree

We show that a graph G on n ⩾ q + 1 vertices (where q ⩾ 2) has the chromatic polynomial P(G;λ) = λ(λ − 1) … (λ − q + 2) (λ − q + 1) 2 (λ − q) n−q−1 if and only if G can be obtained from a q ‐tree T on n vertices by deleting an edge contained in exactly q − 1 triangles of T. Furthermore, we prove that these graphs are triangulated.