A note on local coloring of graphs ?
We study the local k-coloring of graphs. We prove that there exists a graph Gk G k such that ??(Gk)=?(Gk)=k ? ? ( G k ) = ? ( G k ) = k for any k. This result gives a affirmative answer to a problem proposed by Chartrand et al. [2]. We prove that there exists a graph Gk G k with ??(G)=k ? ? ( G ) = k such that every local k -coloring of Gk G k uses all of the colors 1,2,?,k 1 , 2 , ? , k . This result confirms a conjecture proposed by Chartrand et al. [2].
local k-coloring chromatic number local chromatic number