NEIGHBOURHOOD COLORING OF GRAPHS

A graph G has a k-neighbourhood coloring, if there exists a coloring in which for each vertex of G, say v, the sum of the coloring of vertices in N(v) modulo k be non- zero. In this paper we show that every graph G with Δ(G) ≤ 3 has a 3-neighbourhood coloring. We also prove that every tree T has a k-neighbourhood coloring for every k ≥ 3. Moreover, we provide some examples showing that there exists some graphs which do not have k-neighbourhood coloring for some k. Trying to generalize the problem, we proved that for a bipartite graph G of size n and an arbitrary n-dimensional vector we can color each part of G such that sum of colors of at most one of the vertices of each part equals to the corresponding entry of vector.