Relations between the lower domination parameters and the chromatic number of a graph Original Research Article

We establish some new upper bounds for the sum and the product of the domination parameter μ(G), where μ=ir,γ or i, and the chromatic number χ(G) of a graph G. We characterize graphs for which the upper bounds of μ+χ and μχ are achieved. Also an upper bound for the product μχρ is proved for any connected regular graph different from the cycle, where ρ is the packing number. Finally, we give for any graph G with order n an upper bound which is a function of ρ and n for the product of i(G) and i(Ḡ). In particular, this bound improves a result of Cockayne, Favaron, Li and MacGillivray for any graph with a packing number equal to at least three.