The chromaticity of the join of tree and null graph
The chromaticity of the graph G, which is join of the tree Tp and the null graph Oq, is studied. We prove that G is chromatically unique if and only if 1 ≤ p ≤ 3, 1 ≤ q ≤ 2; a graph H and Tp + Op-1 are χ-equivalent if and only if H = T′p + Op-1, where T′p is a tree of order p; H and Tp + Op are χ-equivalent if and only if H ∈ {T′p + Op, T″p+1 + Op-1}, where T′p is a tree of order p, T″p+1 is a tree of order p + 1. We also prove that if p ≤ q, then χ′(G) = ch′(G) = ∆(G); if ∆(G) = |V(G)| - 1, then χ′(G) = ch′(G) = ∆(G) if and only if G ≠ K3.