Discussiones Mathematicae Graph Theory 32(4) (2012) 705-724
doi: 10.7151/dmgt.1636

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Hamiltonian-colored Powers of Strong Digraphs

Garry Johns1, Ryan Jones2,
Kyle Kolasinski2 and Ping Zhang2

1 Saginaw Valley State University
2 Western Michigan University


For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d , the kth power Dk of D is that digraph having vertex set V(D) with the property that (u, v) is an arc of Dk if the directed distance dD(u, v) from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ #x23a1;n/2 #x23a4;, the digraph Dk is Hamiltonian and the lower bound #x23a1;n/2 #x23a4; is sharp. The digraph Dk is distance-colored if each arc (u, v) of Dk is assigned the color i where i = dD(u, v). The digraph Dk is Hamiltonian-colored if Dk contains a properly arc-colored Hamiltonian cycle. The smallest positive integer k for which Dk is Hamiltonian-colored is the Hamiltonian coloring exponent hce(D) of D. For each integer n ≥ 3, the Hamiltonian coloring exponent of the directed cycle Cn of order n is determined whenever this number exists. It is shown for each integer k ≥ 2 that there exists a strong oriented graph Dk such that hce(Dk) = k with the added property that every properly colored Hamiltonian cycle in the kth power of Dk must use all k colors. It is shown for every positive integer p there exists a a connected graph G with two different strong orientations D and D ′ such that hce(D) −hce(D ′) ≥ p.

Keywords: powers of a strong oriented graph, distance-colored digraphs, Hamiltonian-colored digraphs, Hamiltonian coloring exponents

2010 Mathematics Subject Classification: 05C12, 05C15, 05C20, 05C45.


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Received 7 July 2011
Revised 10 December 2011
Accepted 21 December 2011