Vol. 2, No. 1, 2009

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Hamiltonian labelings of graphs

Willem Renzema and Ping Zhang

Vol. 2 (2009), No. 1, 95–114

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 u v 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 k 3, let k be the set of connected graphs G for which there exists a Hamiltonian graph H of order k such that H G cor(H). It is shown that 2k 1 hn(G) k(2k 1) for each G k and that both bounds are sharp.

Hamiltonian labeling, detour distance
Mathematical Subject Classification 2000
Primary: 05C12, 05C45
Secondary: 05C78, 05C15
Received: 21 August 2008
Accepted: 15 November 2008
Published: 18 March 2009

Communicated by Ron Gould
Willem Renzema
Department of Mathematics
Western Michigan University
Kalamazoo, MI 49008
United States
Ping Zhang
Department of Mathematics
Western Michigan University
Kalamazoo, MI 49008
United States