On the maximum number of edges in a graph with prescribed walk-nonrepetitive chromatic number

Abstract

Fix a coloring c: V(G) → N of the vertices of a graph G and let W=v_1 ... v_{2r} be a walk in G. We say that W is repetitive (with respect to c) if c(v_i) = c(v_{i+r}) for i = 1,..., r; and that W is boring if v_i=v_{i+r}, for every i = 1,...,r. Finally, we say that c is a walk-nonrepetitive coloring of G if every repetitive walk is boring, and we denote by σ(G) the walk-nonrepetitive chromatic number, i.e., the minimum number of colors in a walk-nonrepetitive coloring of G. In this paper we explore the maximum number of edges in a graph G with n vertices for which σ(G) = k, for k≥ 4. In [Barát and Wood 2008] it was shown that e(G) ≤ (1/2)(k -1)n. We prove that the corresponding extremal graph is unique. More specifically, we show that e(G) ≤ (1/2)(k -1)n if and only if G is a union of disjoint copies of K_k. We also show that this upper bound can be improved for connected graphs for the case k = 4: if G is a connected graph for which σ(G) = 4 and |V(G)| ≥ 5, then e(G) ≤ (4/3)|V(G)|.