On the Existence of Loose Cycle Tilings and Rainbow Cycles

• Other Authors: Oursler, Roy (Author)
• Format: Doctoral Thesis
• Language: English
• Published: 2019
• Subjects: Mathematics; Degree; Graph; Hypergraph; Loose Cycle; Rainbow Cycle; Tiling
• Online Access: http://hdl.handle.net/2286/R.I.53704

abstract: Extremal graph theory results often provide minimum degree conditions which guarantee a copy of one graph exists within another. A perfect $F$-tiling of a graph $G$ is a collection $\mathcal{F}$ of subgraphs of $G$ such that every element of $\mathcal{F}$ is isomorphic to $F$ and such that every vertex in $G$ is in exactly one element of $\mathcal{F}$. Let $C^{3}_{t}$ denote the loose cycle on $t = 2s$ vertices, the $3$-uniform hypergraph obtained by replacing the edges $e = \{u, v\}$ of a graph cycle $C$ on $s$ vertices with edge triples $\{u, x_e, v\}$, where $x_e$ is uniquely assigned to $e$. This dissertation proves for even $t \geq 6$, that any sufficiently large $3$-uniform hypergraph $H$ on $n \in t \mathbb{Z}$ vertices with minimum $1$-degree $\delta^1(H) \geq {n - 1 \choose 2} - {\Bsize \choose 2} + c(t,n) + 1$, where $c(t,n) \in \{0, 1, 3\}$, contains a perfect $C^{3}_{t}$-tiling. The result is tight, generalizing previous results on $C^3_4$ by Han and Zhao. For an edge colored graph $G$, let the minimum color degree $\delta^c(G)$ be the minimum number of distinctly colored edges incident to a vertex. Call $G$ rainbow if every edge has a unique color. For $\ell \geq 5$, this dissertation proves that any sufficiently large edge colored graph $G$ on $n$ vertices with $\delta^c(G) \geq \frac{n + 1}{2}$ contains a rainbow cycle on $\ell$ vertices. The result is tight for odd $\ell$ and extends previous results for $\ell = 3$. In addition, for even $\ell \geq 4$, this dissertation proves that any sufficiently large edge colored graph $G$ on $n$ vertices with $\delta^c(G) \geq \frac{n + c(\ell)}{3}$, where $c(\ell) \in \{5, 7\}$, contains a rainbow cycle on $\ell$ vertices. The result is tight when $6 \nmid \ell$. As a related result, this dissertation proves for all $\ell \geq 4$, that any sufficiently large oriented graph $D$ on $n$ vertices with $\delta^+(D) \geq \frac{n + 1}{3}$ contains a directed cycle on $\ell$ vertices. This partially generalizes a result by Kelly, K\"uhn, and Osthus that uses minimum semidegree rather than minimum out degree. === Dissertation/Thesis === Doctoral Dissertation Mathematics 2019