A proof of a conjecture on multiset coloring the powers of cycles

For a graph G=(V,E)G=(V,E), let N(u)N(u) denote the set of vertices adjacent to u. A not necessarily proper vertex k-coloring of G is a multiset k-coloring   if M(u)≠M(v)M(u)≠M(v) for every edge uv∈E(G)uv∈E(G), where M(u)M(u) denotes the multiset of colors in N(u)N(u). The minimum k for which G has a multiset k-coloring is the multiset chromatic number  χm(G)χm(G) of G. For positive integers n and r   with 1⩽r

► We give a proof to the conjecture on multiset coloring the powers of cycles.
► We prove that two colors are not enough for multiset coloring Cnr (n⩾3n⩾3, r⩾3r⩾3).
► Frobenius number is used in the proof of the conjecture.