The chromaticity of the join of tree and null graph | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2020. № 50. DOI: 10.17223/20710410/50/7

The chromaticity of the graph G, which is join of the tree T_p and the null graph O_q, is studied. We prove that G is chromatically unique if and only if 1 ≤ p ≤ 3, 1 ≤ q ≤ 2; a graph H and T_p + O_p-1 are χ-equivalent if and only if H = T′_p + O_p-1, where T′_p is a tree of order p; H and T_p + O_p are χ-equivalent if and only if H ∈ {T′_p + O_p, T″_p+1 + O_p-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.