In the present note, exact upper bounds on the sums of chromatic and achromatic numbers of complimentary graphs are determined Which prove in particular a conjecture by Hedetniemi (1966) and imply a bound due to Nordhaus and Gaddum (1956).

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper some variations are considered. First, the sums and products of ψ(G1) and ψ(G2) are examined where G1 ⊕ G2 = K(s, s), and ψ is the independence, domination, or independent domination number, inter alia. In particular, it is shown that the maximum valu...

In this paper, Nordhaus-Gaddum type relations on open support independence number of some derived graphs path related under addition and multiplication are studied.

For a graph G on n vertices with chromatic number χ(G), the Nordhaus–Gaddum inequalities state that d2 √ ne ≤ χ(G) + χ(G) ≤ n + 1, and n ≤ χ(G) · χ(G) ≤ ⌊( n+1 2 )2⌋ . Much analysis has been done to derive similar inequalities for other graph parameters, all of which are integer-valued. We determine here the optimal Nordhaus–Gaddum inequalities for the circular chromatic number and the fraction...

Nordhaus and Gaddum proved, for any graph G, that χ(G) + χ(G) 6 n + 1, where χ is the chromatic number and n = |V (G)|. Finck characterized the class of graphs, which we call NG-graphs, that satisfy equality in this bound. In this paper, we provide a new characterization of NG-graphs, based on vertex degrees, which yields a new polynomial-time recognition algorithm and efficient computation of ...

We establish the bounds 43 6 bν 6 bξ 6 √ 2, where bν and bξ are the NordhausGaddum sum upper bound multipliers, i.e., ν(G)+ν(G) 6 bν |G| and ξ(G)+ξ(G) 6 bξ|G| for all graphs G, and ν and ξ are Colin de Verdière type graph parameters. The Nordhaus-Gaddum sum lower bound for ν and ξ is conjectured to be |G| − 2, and if these parameters are replaced by the maximum nullity M(G), this bound is calle...