On q-graphs Original Research Article

Let q be a positive integer. A graph G is said to be a q-graph if it contains a q-tree as one of its spanning subgraphs. A connected graph is a 1-graph. (1) We prove that if G is a q-graph with |V(G)| > q, then the multiplicity of the root q of the chromatic polynomial P(G, λ) is the number of q-blocks (maximal subgraphs without any separating Kq in G). This is a generalization of a result in Whitehead and Zhao (1984). (2) We give a characterization of G being chromatically unique. This is a generalization of a result in Chia (1986). (3) Let q ⩽ m and Km(q) be the graph obtained by joining q edges between a complete graph Km with m vertices and a vertex. We show that Km(q) is chromatically unique. This is a generalization of a result in Giudici (1985).