On Irregular Colorings of Graphs

For a graph G and a proper coloring c : V (G) → {1, 2, . . . , k} of the vertices of G for some positive integer k , the color code of a vertex v of G (with respect to c ) is the ordered (k + 1) -tuple code(v) = (a0, a1, . . . , ak) where a0 is the color assigned to v and for 1 ≤ i ≤ k , ai is the number of vertices adjacent to v that are colored i . The coloring c is irregular if distinct vertices have distinct color codes and the irregular chromatic number χir(G) of G is the minimum positive integer k for which G has an irregular k -coloring. We establish sharp upper and lower bounds for the irregular chromatic number of a disconnected graph in terms of the irregular chromatic numbers of its components. Irregular chromatic numbers of some classes of disconnected graphs are determined. It is shown that if G is a nontrivial graph of order n , then 2 √ n ≤ χir(G) + χir(G) ≤ 2n , n ≤ χir(G)χir(G) ≤ n , and each bound in these inequalities is sharp.