On the structure and chromaticity of graphs in which any two colour classes induce a tree Original Research Article

For r⩾2, let Fr be the family of graphs G which possesses an independent set partition {A1,h.,Ar} such that the subgraph induced by Ai ⌣ Aj in G is a tree for all i and j with 1 ⩽ i < j ⩽ r. Let v(G) and t(G) denote, respectively, the order of G and the number of triangles in G. Define t(r,n) = max{t(G) | G ∈ Fr, v(G) = n}. It is known that the family {G | G ∈ Fr,t(G) = t(r,n)} is the family of (r − 1)-trees of order n, and the family of r-trees of the same order forms a chromatically equivalence class. In this paper, we determine the structure of the graphs in the family {G | G ∈ Fr, t(G) = t(r,n) − 1} and apply the structural results to investigate the chromaticity of the graphs of the family. A number of new families of chromatically unique graphs are discovered.