This Is AuburnElectronic Theses and Dissertations

Forbidding Monchromatic and Rainbow Cycles and Families of Cycles

Date

2023-05-01

Author

DeMars, Derrick

Type of Degree

PhD Dissertation

Department

Mathematics and Statistics

Abstract

In this dissertation, we avoid certain cycles and families of cycles in complete graphs. Introduced by Axenovich and Choi \cite{2011NoteMonotonicity}, the mixed Ramsey spectrum, $\mrs{F}{H}$, is the set of numbers $k$ such that for some $k$-edge coloring of $K_{n}$ there is neither a monochromatic copy of $F\subseteq K_{n}$ nor a rainbow copy of $H\subseteq K_{n}$. The values for the following spectrums are shown. Let $m$ and $n$ be an integers. For $n>1$, $\mrs{K_3}{K_3}$=$\left\{ g\left(n\right),\dots,n-1\right\} $, in which $g\left(n\right)\in\left\{ \left\lceil 2\log_{5}n\right\rceil ,\left\lceil 2\log_{5}n\right\rceil +1\right\}$. For all $m$ and $n$, where $3\leq m\leq n$, $\left\{ n+2-m,\dots,n+1-m+\binom{m-1}{2}\right\} \subseteq \mrs{C_m}{C_m}$. For $n\geq4$, $\max\mrs{C_4}{C_4}=n$. Note: $\max\mrs{C_4}{C_4}=n$ was a result shown by Axenovich and Choi \cite{2010AvichChoifixedmonosubgraphs}, we provide an alternate proof. We extended the definition of the mixed Ramsey spectrum from graphs to families of graphs. For families of graphs $\mathcal{F},\mathcal{H}$, $\mrs{ \mathcal{F} }{ \mathcal{H} }$ is the set of numbers $k$ such that for some $k$-edge coloring of $K_{n}$, there is no monochromatic copy of any $F\subseteq\mathcal{F}$ in $K_{n}$ nor any rainbow copy of any $H\subseteq\mathcal{H}$ in $K_{n}$. It is shown that for all $n\geq2$, \\$\mrs{\left\lbrace \text{odd cycles}\right\rbrace }{\left\lbrace \text{cycles}\right\rbrace }$$=\left\{ \left\lceil \log_{2}n\right\rceil ,\dots,n-1\right\}$.