A note on local coloring of graphs

A local k-coloring of a graph G is a function f:V(G)→{1,2,⋯,k} such that for each S⊆V(G), 2≤|S|≤3, there exist u,v∈S with |f(u)−f(v)| at least the size of the subgraph induced by S. The local chromatic number of G is χℓ(G)=min⁡{k:G has a local k-coloring}. Chartrand et al. [2] asked: does there exist a graph Gk such that χℓ(Gk)=χ(Gk)=k? Furthermore, they conjectured that for every positive integer k, there exists a graph Gk with χℓ(G)=k such that every local k-coloring of Gk uses all of the colors 1,2,⋯,k. In this paper we give a affirmative answer to the problem and confirm the conjecture.