#### Graduation Year

2015

#### Abstract

A *k*-distant tree consists of a main path, called the *spine*, such that each vertex on the spine is joined by an edge to an end-vertex of at most one path on at most *k* vertices. Those paths, along with the edge joining them to the spine, are called *tails*. When every vertex on the spine has exactly one incident tail of length *k* we call the tree a *uniform k*-distant tree. We show that every uniform *k*-distant tree admits both a graceful- and an *α*-labeling.

For a graph *G* and a positive integer *a*, define *app _{a}*(

*G*) to be the graph obtained from appending

*a*leaves to each leaf in

*G*. When

*G*is a uniform

*k*-distant tree, we show that

*app*(

_{a}*G*) admits both a graceful- and an

*α*-labeling.

#### Disciplines

Discrete Mathematics and Combinatorics | Mathematics

#### Recommended Citation

Wenger Diller, Kimberly, "Two Rosa-type Labelings of Uniform k-distant Trees and a New Class of Trees" (2015). *Honors Projects.* Paper 17.

http://digitalcommons.iwu.edu/math_honproj/17

## Comments

Faculty Advisor, Dr. Dan Roberts