Local coloring of self complementary graphs

摘要

Let G = ( V , E ) be a graph. A local coloring of a graph G of order at least 2 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 size 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 ? ( c ) . The local chromatic number of G is χ ? ( G ) = min { χ ? ( c ) } , where the minimum is taken over all local colorings c of G . In this paper we study the local coloring for some self complementary graphs. Also we present a sc-graph with local chromatic number k for any given integer k ≥ 6 .