Minimal coloring and strength of graphs

Abstract

Let G be a graph. A minimal coloring of G is a coloring which has the smallest possible sum among all proper colorings of G, using natural numbers. The vertex-strength of G, denoted by s( G), is the minimum number of colors which is necessary to obtain a minimal coloring. In this note we study these concepts, and define a new concept called the edge-strength of G, denoted by s′( G). We prove the celebrated Brooks’ theorem for χ( G) replaced by s( G) and we also prove the following upper bound for s( G): s(G)⩽ col(G)+Δ(G) 2 , where col( G) is an invariant based on linear orderings of the vertices. Also, it is proved that s′( G) lies between Δ( G) and Δ( G)+1, as for χ′( G), but it may be not equal to χ′( G). Based on our results about vertex-strength we conjecture s(G)⩽ χ(G)+Δ(G) 2 .