On the Nash number and the diminishing Grundy number of a graph

A Nash k-colouring is a (S1,…,Sk) such that every vertex of Si adjacent to in Sj, whenever |Sj|≥|Si|. The Number G, denoted by Nn(G), the largest k G admits k-colouring. diminishing greedy that, for all 1≤j0 χ(G)≤εω(G)+(1−ε)(Δ(G)+1). then trees forests, Γ(F)−1≤Nn(F)≤Γ↓(F)≤Γ(F). Finally complexity related problems. computing NP-hard even when input bipartite chordal. deciding whether satisfies γ1(G)=γ2(G) pair (γ1,γ2) with γ1∈{Nn,Γ↓} γ2∈{ω,χ,Γ,Δ+1}.