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 .