#### Date of Award

6-2018

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Ping Zhang

#### Second Advisor

Dr. Gary Chartrand

#### Third Advisor

Dr. Clifton Ealy

#### Fourth Advisor

Dr. Ebrahim Salehi

#### Abstract

For a graph *G *of size *m*, a graceful labeling of *G *is an injective function *f *: *V *(*G*) *→*

*{*0*,** *1*,** **.** **.** **.** **,** **m**}** *that gives rise to a bijective function *f** *^{1}* *: *E*(*G*) *→** {*1*,** *2*,** **.** **.** **.** **,** **m**}** *defined by *f** ** ^{1}*(

*uv*) =

*|*

*f*

*(*

*u*)

*−*

*f*

*(*

*v*)

*|*. A graph is graceful if it has a graceful labeling. Over the years, a number of variations of graceful labelings have been introduced, some of which have been described in terms of colorings.

A proper (vertex) coloring of a graph *G *is an assignment of colors to the vertices of *G *such that adjacent vertices are assigned distinct colors. The minimum number of colors required of a proper vertex coloring of *G *is its chromatic number, *χ*(*G*). Similarly, a proper edge coloring of a graph *G *is an assignment of colors to the edges of *G *such that adjacent edges are assigned distinct colors. The minimum number of colors required of a proper edge coloring of *G *is its chromatic index, *χ** ^{l}*(

*G*).

A proper vertex coloring *c *: *V *(*G*) → {1*, *2*, . . . , k*} is called a graceful *k*-coloring

if the induced edge coloring *c*^{l}* *defined by *c** ^{l}*(

*uv*) = |

*c*(

*u*) −

*c*(

*v*)| is also proper. The

minimum positive integer *k *for which *G *has a graceful *k*-coloring is its graceful chromatic number *χ** _{g}*(

*G*). These chromatic numbers are determined for several well-known classes of graphs, including cycles, wheels and caterpillars. An upper bound for the graceful chromatic number of trees is determined in terms of its maximum degree. We also present several other results and conjectures on this coloring concept.

A graph is edge-colored if each of its edges is assigned a color (where adjacent edges may be assigned the same color). Let *G *be an edge-colored connected graph. A path *P *in an edge-colored graph *G *is a rainbow path of *G *if no two edges of *P *are colored the same. An edge coloring *c *of a connected graph *G *is a rainbow coloring of *G *if every pair of distinct vertices of *G *are connected by a rainbow path in *G*. In this case, *G *is rainbow-connected. The minimum number of colors needed for a rainbow coloring of *G *is referred to as the rainbow connection number of *G *and is denoted by rc(*G*). Analogously, we say that a path *P *in an edge-colored connected graph *G *is a proper path in *G *if no two adjacent edges of *P *are colored the same. An edge coloring *c *of a connected graph *G *is a proper-path coloring of *G *if every pair of distinct vertices of *G *are connected by a proper path in *G*. If *k *colors are used, then *c *is referred to as a proper-path k-coloring. The minimum *k *for which *G *has a proper-path *k*-coloring is called the proper connection number pc(*G*) of *G*. Since rainbow and proper connection numbers were introduced in 2006 and 2009, respectively, these numbers have been studied by many researchers with a wide variety of applications.

A graph *G *is Hamiltonian-connected if every pair of vertices of *G *are connected by a Hamiltonian path, that is, every pair of vertices of *G *are connected by a path containing every vertex of *G*. An edge coloring of a Hamiltonian-connected graph *G *is a Hamiltonian-connected rainbow coloring if every two vertices of *G *are connected by a rainbow Hamiltonian path. The minimum number of colors required of a Hamiltonian- connected rainbow coloring of *G *is the rainbow Hamiltonian-connection number hrc(*G*) of *G*. If *G *has order *n *and size *m*, then *n *− 1 ≤ hrc(*G*) ≤ *m*. The rainbow Hamiltonian- connection number is investigated for the Cartesian product of complete graphs and of odd cycles with *K*_{2}. As a result of this, both the lower bound *n *− 1 and the upper bound *m *for hrc(*G*) are shown to be sharp. Furthermore, the rainbow Hamiltonian- connection numbers are investigated for several classes of Hamiltonian-connected graphs, including the join of graphs *G *and *K*_{2}, where *G *contains a Hamiltonian path, squares of Hamiltonian graphs, and Hamiltonian graphs of minimum size. Correspondingly, an edge coloring of a Hamiltonian-connected graph *G *is a proper Hamiltonian-path coloring if every two vertices of *G *are connected by a properly colored Hamiltonian path. The minimum number of colors in a proper Hamiltonian-path coloring of *G *is the proper Hamiltonian-connection number of *G*, denoted by hpc(*G*). Proper Hamiltonian- connection numbers are determined for several classes of Hamiltonian-connected graphs as well as two classes of Hamiltonian-connected graphs of minimum size. In particular, it is shown that hpc(*K** _{n}*) = 2 for

*n*≥ 4 and hpc(

*C*□

*K*

_{2}) = 3 for all prisms

*C*□

*K*

_{2}, where

*C*is an odd cycle.

Let *G *be an edge-colored connected graph, where adjacent edges may be colored the same, and let *f *be the length of a longest path in *G*. For an integer *k *≥ 2, a path *P *in *G *is a *k*-rainbow path if every subpath of *P *having length at most *k *is a rainbow path. In particular, every *k*-rainbow path of length at most *k *is a rainbow path. An edge coloring *c *of *G *is a *k*-rainbow coloring if every pair of distinct vertices of *G *are connected by a *k*-rainbow path in *G*. In this case, the graph *G *is *k*-rainbow connected (with respect to *c*). If *j *colors are used to produce a *k*-rainbow coloring of *G*, then *c *is referred to as a *k*-rainbow *j*-edge coloring (or simply a *k*-rainbow *j*-coloring). The minimum *j *for which *G *has a *k*-rainbow *j*-coloring is called the *k*-rainbow connection number rc* _{k}*(

*G*) of

*G*. Hence, we have rc

_{2}(

*G*) = pc(

*G*), rc(

*G*) = rc(

*G*) if

*£*is the length of a longest path in

*G*, and

*k*-rainbow colorings are intermediate to rainbow and proper-path colorings for all integers

*k*with 2 ≤

*k*≤ ℓ. Therefore, for a nontrivial connected graph

*G*of size

*m*whose longest paths have length ℓ,

1 ≤ pc(*G*) = rc_{2}(*G*) ≤ rc_{3}(*G*) ≤ · · · ≤ rc(*G*) = rc(*G*) ≤ *m.*

For an integer *k *≥ 2, a Hamiltonian path *P *in *G *is a *k*-rainbow Hamiltonian path if every subpath of *P *having length at most *k *is a rainbow path. An edge coloring of *G *is a *k*-rainbow Hamiltonian-path coloring if every two vertices of *G *are connected by a *k*-rainbow Hamiltonian path in *G*. The minimum number of colors in a *k*-rainbow Hamiltonian-path coloring of *G *is the *k*-rainbow Hamiltonian-connection number of *G*. We investigate the *k*-rainbow Hamiltonian-path colorings in two well-known classes of Hamiltonian-connected graphs, namely the join *G *∨ *K*_{1} of a Hamiltonian graph *G *and the trivial graph *K*_{1} and the prism *G *□ *K*_{2} where *G *is a Hamiltonian graph of odd order. Other results and open questions are also presented.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Byers, Alexis D., "Graceful Colorings and Connection in Graphs" (2018). *Dissertations*. 3308.

https://scholarworks.wmich.edu/dissertations/3308