Involve: A Journal of Mathematics

  • Involve
  • Volume 2, Number 1 (2009), 95-114.

Hamiltonian labelings of graphs

Willem Renzema and Ping Zhang

Full-text: Open access

Abstract

For a connected graph G of order n, the detour distance D(u,v) between two vertices u and v in G is the length of a longest uv path in G. A Hamiltonian labeling of G is a function c:V(G) such that |c(u)c(v)|+D(u,v)n for every two distinct vertices u and v of G. The value hn(c) of a Hamiltonian labeling c of G is the maximum label (functional value) assigned to a vertex of G by c; while the Hamiltonian labeling number hn(G) of G is the minimum value of Hamiltonian labelings of G. Hamiltonian labeling numbers of some well-known classes of graphs are determined. Sharp upper and lower bounds are established for the Hamiltonian labeling number of a connected graph. The corona cor(F) of a graph F is the graph obtained from F by adding exactly one pendant edge at each vertex of F. For each integer k3, let k be the set of connected graphs G for which there exists a Hamiltonian graph H of order k such that HG cor(H). It is shown that 2k1 hn(G)k(2k1) for each Gk and that both bounds are sharp.

Article information

Source
Involve, Volume 2, Number 1 (2009), 95-114.

Dates
Received: 21 August 2008
Accepted: 15 November 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513799120

Digital Object Identifier
doi:10.2140/involve.2009.2.95

Mathematical Reviews number (MathSciNet)
MR2501348

Zentralblatt MATH identifier
1167.05024

Subjects
Primary: 05C12: Distance in graphs 05C45: Eulerian and Hamiltonian graphs
Secondary: 05C78: Graph labelling (graceful graphs, bandwidth, etc.) 05C15: Coloring of graphs and hypergraphs

Keywords
Hamiltonian labeling detour distance

Citation

Renzema, Willem; Zhang, Ping. Hamiltonian labelings of graphs. Involve 2 (2009), no. 1, 95--114. doi:10.2140/involve.2009.2.95. https://projecteuclid.org/euclid.involve/1513799120


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References

  • G. Chartrand and P. Zhang, Chromatic graph theory, Chapman & Hall/CRC, Boca Raton, FL, 2008.
  • G. Chartrand, D. Erwin, P. Zhang, and F. Harary, “Radio labelings of graphs”, Bull. Inst. Combin. Appl. 33 (2001), 77–85.
  • G. Chartrand, D. Erwin, and P. Zhang, “Radio antipodal colorings of graphs”, Math. Bohem. 127:1 (2002), 57–69.
  • G. Chartrand, L. Nebeský, and P. Zhang, “Bounds for the Hamiltonian chromatic number of a graph”, pp. 113–125 in Proceedings of the Thirty-third Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 2002), vol. 157, 2002.
  • G. Chartrand, L. Nebeský, and P. Zhang, “Hamiltonian colorings of graphs”, Discrete Appl. Math. 146:3 (2005), 257–272.
  • G. Chartrand, L. Nebesky, and P. Zhang, “On Hamiltonian colorings of graphs”, Discrete Math. 290:2-3 (2005), 133–143.
  • L. Nebeský, “Hamiltonian colorings of graphs with long cycles”, Math. Bohem. 128:3 (2003), 263–275.
  • L. Nebeský, “The Hamiltonian chromatic number of a connected graph without large Hamiltonian-connected subgraphs”, Czech. Math. J. 56:2 (2006), 317–338.