Involve: A Journal of Mathematics

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

Hamiltonian labelings of graphs

Willem Renzema and Ping Zhang

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Hamiltonian labeling detour distance


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

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