Date of Defense
Ping Zhang, Mathematics
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.
Renzema, Willem, "Hamiltonian Labelings of Graphs" (2007). Honors Theses. 295.
Honors Thesis-Campus Only