An evaluation on the gracefulness and colouring of graphs

In this thesis we shall introduce two interesting topics from graph theory and begin to explore what happens when we combine these together. We will be focusing on an area known as graph colouring and assessing it alongside a very unique set of graphs called graceful graphs. The two topic areas, although not mixed together often, nicely support each other in introducing various findings from each of the topics. We will start by investigating graceful graphs and determining what classes of graph can be deemed to be graceful, before introducing some of the fundamentals of graph colouring. Following this we can then begin to investigate the two topics combined and will see a whole range of results, including some fascinating less well known discoveries. Furthermore, we will introduce some different types of graph colouring based off the properties of graceful graphs. Later in the thesis there will also be a focus on tree graphs, as they have had a huge influence on research involving graceful graphs over the years. We will then conclude by investigating some results that have been formulated by combining graceful graphs with a type of graph colouring known as total colouring

On d-graceful labelings

In this paper we introduce a generalization of the well known concept of a graceful labeling. Given a graph G with e=dm edges, we call d-graceful labeling of G an injective function from V(G) to the set {0,1,2,..., d(m+1)-1} such that {|f(x)-f(y)| | [x,y]\in E(G)} ={1,2,3,...,d(m+1)-1}-{m+1,2(m+1),...,(d-1)(m+1)}. In the case of d=1 and of d=e we find the classical notion of a graceful labeling and of an odd graceful labeling, respectively. Also, we call d-graceful \alpha-labeling of a bipartite graph G a d-graceful labeling of G with the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other one. We show that these new concepts allow to obtain certain cyclic graph decompositions. We investigate the existence of d-graceful \alpha-labelings for several classes of bipartite graphs, completely solving the problem for paths and stars and giving partial results about cycles of even length and ladders.Comment: In press on Ars Combi

SOME CARTESIAN PRODUCTS OF A PATH AND PRISM RELATED GRAPHS THAT ARE EDGE ODD GRACEFUL

Let $G$ be a connected undirected simple graph of size $q$ and let $k$ be the maximum number of its order and its size. Let $f$ be a bijective edge labeling which codomain is the set of odd integers from 1 up to $2q-1$. Then $f$ is called an edge odd graceful on $G$ if the weights of all vertices are distinct, where the weight of a vertex $v$ is defined as the sum $mod(2k)$ of all labels of edges incident to $v$. Any graph that admits an edge odd graceful labeling is called an edge odd graceful graph. In this paper, some new graph classes that are edge odd graceful are presented, namely some cartesian products of path of length two and some circular related graphs

On the graceful polynomials of a graph

Every graph can be associated with a family of homogeneous polynomials, one for every degree, having as many variables as the number of vertices. These polynomials are related to graceful labellings: a graceful polynomial with all even coefficients is a basic tool, in some cases, for proving that a graph is non-graceful, and for generating a possibly infinite class of non-graceful graphs. Graceful polynomials also seem interesting in their own right. In this paper we classify graphs whose graceful polynomial has all even coefficients, for small degrees up to 4. We also obtain some new examples of non-graceful graphs

A Study on Integer Additive Set-Graceful Graphs

A set-labeling of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$, where $X$ is a finite set and a set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus}:E(G)\rightarrow \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus}f(v)$ for every $uv{\in} E(G)$ is also injective. An integer additive set-labeling is an injective function $f:V(G)\rightarrow \mathcal{P}(\mathbb{N}_0)$, $\mathbb{N}_0$ is the set of all non-negative integers and an integer additive set-indexer is an integer additive set-labeling such that the induced function $f^+:E(G) \rightarrow \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. In this paper, we extend the concepts of set-graceful labeling to integer additive set-labelings of graphs and provide some results on them.Comment: 11 pages, submitted to JARP

Beyond Problem-Solving: Elementary Students’ Mathematical Dispositions When Faced With The Challenge Of Unsolved Problems

The goal of this study was to document the characteristics of students’ dispositions towards mathematics when they engaged in the exploration of parts of unsolved problems: Graceful Tree Conjecture and Collatz Conjecture. Ten students, Grades 4 and 5, from an after-school program in the Midwest participated in the study. I focused on the cognitive, affective, and conative aspects of their mathematical dispositions as they participated in 7 problem-solving sessions and two interviews. With regard to cognitive aspects of the students’ dispositions, I focused on the students attempts to identify and justify patterns for labeling graphs. Overall, the unsolved problems were accessible to the students and they found patterns that enabled them to gracefully label specific classes of graphs for the Graceful Tree Conjecture. With regard to affective aspects of students’ dispositions, I found five themes that characterized their beliefs about the nature of mathematics. Also, students exhibited a variety of emotions throughout the study. The two emotions exhibited most frequently were frustration and joy. The third type of disposition that students exhibited was the conative construct of perseverance. This was related to the interplay of frustration and joy and characterized the productive struggle that students experienced throughout the study. To examine students’ dispositions in greater depth, I conducted a case study analysis of the positional identities of two students, which I report in a detailed narrative