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

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Abstract

For a graph G=(V,E), let 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) for every edge uvE(G), where M(u) denotes the multiset of colors in N(u). The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For positive integers n and r with 1r<n, let Cnr denote the r-th power of cycle Cn. It was conjectured in Chartrand et al. (2009) [2] that for every integer r3, there exists an integer f(r) such that χm(Cnr)=3 for all nf(r). This paper gives an affirmative answer to this conjecture.

Highlights

► 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 (n3, r3). ► Frobenius number is used in the proof of the conjecture.

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Project 10971025 supported by NSFC.

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