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| 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. A vertex coloring χ : V (G) → [k] of a graph G is said to be proper if no pair of adjacent vertices inG 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) 6= 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 exactly the irregular chromatic number of Cn and Pn for n ≥ 3, thus verifying two conjectures given by Okamoto, Radcliffe and Zhang.