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

Abstract

A Nashk-colouring is a k-colouring (S1,…,Sk) such that every vertex of Si is adjacent to a vertex in Sj, whenever |Sj|≥|Si|. The Nash Number of G, denoted by Nn(G), is the largest k such that G admits a Nash k-colouring. A diminishing greedyk-colouring is a k-colouring (S1,…,Sk) such that, for all 1≤j<i≤k, |Si|≤|Sj| and every vertex of Si is adjacent to a vertex in Sj. The diminishing Grundy Number of G, denoted by Γ↓(G), is the largest k such that G admits a diminishing greedy k-colouring. In this paper, we prove some properties of Nn and Γ↓. We mainly study the relations between them and other graph parameters such as the clique number ω, the chromatic number χ, the Grundy number Γ, and the maximum degree Δ. In particular we study the chain of inequalities ω(G)≤χ(G)≤Nn(G)≤Γ↓(G)≤Γ(G)≤Δ(G)+1. We show each inequality γ1(G)≤γ2(G) of this chain is loose, in the sense that there is no function f such that γ2(G)≤f(γ1(G)). We also prove the existence or non-existence of inequalities analogue to Reed’s one stating that there exists ε>0 such that χ(G)≤εω(G)+(1−ε)(Δ(G)+1).We then study the Nash number and the diminishing Grundy number of trees and forests, and prove that Γ(F)−1≤Nn(F)≤Γ↓(F)≤Γ(F).Finally we study the complexity of related problems. We show that computing the Nash number or the diminishing Grundy number is NP-hard even when the input graph is bipartite or chordal. We also show that deciding whether a graph satisfies γ1(G)=γ2(G) is NP-hard for every pair (γ1,γ2) with γ1∈{Nn,Γ↓} and γ2∈{ω,χ,Γ,Δ+1}.