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## Discussiones Mathematicae Graph Theory

2012 | 32 | 4 | 705-724
Tytuł artykułu

### Hamiltonian-colored powers of strong digraphs

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power $D^k$ of D is that digraph having vertex set V(D) with the property that (u, v) is an arc of $D^k$ if the directed distance $^{→}d_D(u,v)$ from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph $D^k$ is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph $D^k$ is distance-colored if each arc (u, v) of $D^k$ is assigned the color i where $i = ^{→}d_D(u,v)$. The digraph $D^k$ is Hamiltonian-colored if $D^k$ contains a properly arc-colored Hamiltonian cycle. The smallest positive integer k for which $D^k$ 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.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
705-724
Opis fizyczny
Daty
wydano
2012
otrzymano
2011-07-07
poprawiono
2011-12-10
zaakceptowano
2011-12-21
Twórcy
autor
• Saginaw Valley State University
autor
• Western Michigan University
autor
• Western Michigan University
autor
• Western Michigan University
Bibliografia
• [1] G. Chartrand, R. Jones, K. Kolasinski and P. Zhang, On the Hamiltonicity of distance-colored graphs, Congr. Numer. 202 (2010) 195-209.
• [2] G. Chartrand, K. Kolasinski and P. Zhang, The colored bridges problem, Geographical Analysis 43 (2011) 370-382, doi: 10.1111/j.1538-4632.2011.00827.x.
• [3] G. Chartrand, L. Lesniak and P. Zhang, Graphs & Digraphs, Fifth Edition (Chapman & Hall/CRC, Boca Raton, FL, 2011).
• [4] H. Fleischner, The square of every nonseparable graph is Hamiltonian, Bull. Amer. Math. Soc. 77 (1971) 1052-1054, doi: 10.1090/S0002-9904-1971-12860-4.
• [5] A. Ghouila-Houri, Une condition suffisante d'existence d'un circuit Hamiltonien, C. R. Acad. Sci. Paris 251 (1960) 495-497.
• [6] R. Jones, K. Kolasinski and P. Zhang, On Hamiltonian-colored graphs, Util. Math. to appear.
• [7] M. Sekanina, On an ordering of the set of vertices of a connected graph, Publ. Fac. Sci. Univ. Brno 412 (1960) 137-142.
Typ dokumentu
Bibliografia
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