deBruijn-like sequences and the irregular chromatic number of paths and cycles

A deBruijn sequence of order k , or a k -deBruijn sequence, over an alphabet A is a sequence of length | A | k in which the last element is considered adjacent to the first and every possible k -tuple from A appears exactly once as a string of k -consecutive elements in the sequence. We will say that a cyclic sequence is deBruijn-like if for some k , each of the consecutive k -element substrings is distinct. ex coloring χ : V ( G ) → [ k ] of a graph G is said to be proper if no pair of adjacent vertices in G receive the same color. Let C ( v ; χ ) denote the multiset of colors assigned by a coloring χ to the neighbors of vertex v . A proper coloring χ of G is irregular if χ ( u ) = χ ( v ) implies that C ( u ; χ ) ≠ C ( v ; χ ) . The minimum number of colors needed to irregularly color G is called the irregular chromatic number of G . The notion of the irregular chromatic number pairs nicely with other parameters aimed at distinguishing the vertices of a graph. In this paper, we demonstrate a connection between the irregular chromatic number of cycles and the existence of certain deBruijn-like sequences. We then determine the exact irregular chromatic number of C n and P n for n ≥ 3 , thus verifying two conjectures given by Okamoto, Radcliffe and Zhang.