Local colourings of Cartesian product graphs

A local colouring 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 number of edges in the subgraph induced by S. The maximum colour assigned by c is the value χℓ(c) of c, and the local chromatic number of G is χℓ(G)=min {χℓ(c): c is a local colouring of G}. In this note the local chromatic number is determined for Cartesian products G □ H, where G and GH are 3-colourable graphs. This result in part corrects an error from Omoomi and Pourmiri [On the local colourings of graphs, Ars Combin. 86 (2008), pp. 147–159]. It is also proved that if G and H are graphs such that χ(G)≤⌊ χℓ(H)/2 ⌋, then χℓ(G □ H)≤χℓ(H)+1.