Local coloring of Kneser graphs

Abstract

A local coloring of a graph G is a function c : V ( G ) → N having the property that for each set S ⊆ V ( G ) with 2 ≤ | S | ≤ 3 , there exist vertices u , v ∈ S such that | c ( u ) − c ( v ) | ≥ m S , where m S is the number of edges of the induced subgraph 〈 S 〉 . The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ ℓ ( c ) . The local chromatic number of G is χ ℓ ( G ) = min { χ ℓ ( c ) } , where the minimum is taken over all local colorings c of G . The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207–221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K ( n , k ) for some values of n and k is determined.