On local colorings of Cartesian product graphs

A local coloring of a graph G is a function c : V (G) → N such that for each S ⊆ V (G), 2 ≤ |S| ≤ 3, there exist u, v ∈ S with |c(u) − c(v)| at least the size of the subgraph induced by S. The maximum color assigned by c is the value χl(c) of c, and the local chromatic number of G is χl(G) = min{χl(c) : c local coloring of G}. In this note the local chromatic number is determined for Cartesian products of 3-chromatic graphs which in part corrects an error from [Omoomi, Pourmiri, On the local colorings of graphs, Ars Combin. 86 (2008) 147–159]. It is also proved that if G and H are graphs such that χ(G) ≤ ⌊χl(H)/2⌋ and H is triangle-free, then χl(G H) = χl(H).