Date of Defense

2007

Department

Mathematics

First Advisor

Ping Zhang, Mathematics

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 the longest u - v path in G. A Hamiltonian labeling of G is a function of c:V(G)→ℕ such that |c(u)-c(v)|+D(u,v)≥n for every two distinct vertices of 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 a Hamiltonian labeling of G. Hamiltonian labeling numbers of several well-known classes of graphs are determined. The corona cor(G) of a graph G is the graph obtained from G by adding exactly one pendant edge at each vertex of G. For each integer k ≥ 3 let Hk 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 ∈ Hk and both bounds are sharp. It is also shown that if G is a nontrivial connected graph of order n, then n ≤ hn(G) ≤ n + (n-2)2. Furthermore, if G is Hamiltonian, then hn(G)=n, while hn(G)=n+(n-2)2 if and only if G is a star. Upper and lower bounds for the Hamiltonian labeling number of a connected graph are established in terms of the order and diameter of the graph as well as other graphical parameters.

Access Setting

Honors Thesis-Campus Only

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