Decomposing 8-regular graphs into paths of length 4

A $T$-decomposition of a graph $G$ is a set of edge-disjoint copies of $T$ in $G$ that cover the edge set of $G$. Graham and H\"aggkvist (1989) conjectured that any $2\ell$-regular graph $G$ admits a $T$-decomposition if $T$ is a tree with $\ell$ edges. Kouider and Lonc (1999) conjectured that, in the special case where $T$ is the path with $\ell$ edges, $G$ admits a $T$-decomposition $\mathcal{D}$ where every vertex of $G$ is the end-vertex of exactly two paths of $\mathcal{D}$, and proved that this statement holds when $G$ has girth at least $(\ell+3)/2$. In this paper we verify Kouider and Lonc's Conjecture for paths of length $4$

Some Topics of Special Interest in Graph Theory

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Discussion On Some Important Topics in Graph Theory

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Some Investigations in the Theory of Graphs

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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 Properties of r\u3csub\u3ew\u3c/sub\u3e-Regular Graphs

If every vertex in a graph G has the same degree, then the graph is called a regular graph. That is, if deg(v) = r for all vertices in the graph, then it is denoted as an r-regular graph. A graph G is said to be vertex-weighted if all of the vertices are assigned weights. A generalized definition for degree regularity for vertex-weighted graphs can be stated as follows: A vertex-weighted graph is said to be rw-regular if the sum of the weights in the neighborhood of every vertex is rw. If all vertices are assigned the unit weight of 1, then this is equivalent to the definition for r-regular graphs. In this thesis, we determine if a graph has a weighting scheme that makes it a weighted regular graph or prove no such scheme exists for a number of special classes of graphs such as paths, stars, caterpillars, spiders and wheels