#### 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 *H _{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 2

*k*- 1 ≤ hn(

*G*) ≤

*k*(2

*k*-1) for each

*G*∈

*H*and both bounds are sharp. It is also shown that if

_{k}*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.

#### Recommended Citation

Renzema, Willem, "Hamiltonian Labelings of Graphs" (2007). *Honors Theses*. 295.

http://scholarworks.wmich.edu/honors_theses/295

#### Access Setting

Honors Thesis-Campus Only