\relax \citation{AchlioptasMolloy} \citation{AchlioptasMolloy} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.1}{The Achlioptas Molloy Paper}}{1}} \newlabel{sect:ERGraphs}{{1.2}{1}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.2}{The Erd\"{o}s-R\'{e}nyi Graph}}{1}} \citation{KnuthStableMariagesBook} \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Model Description}}{2}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A Visualization for the List Counts: At time $t$, the algorithm is described by the counts $X_1(t)$, $X_2(t)$, $X_3(t)$, $Y_1(t)$, $Y_2(t)$, $Y_3(t)$. If a vertex does not have possible coloring, the algorithm fails.}}{3}} \newlabel{fig1}{{1}{3}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{The Simplified Model}}{4}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{The Distribution of $N_t$ and $S_t$}}{5}} \newlabel{sect:balance}{{3.2}{5}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{$X_1(t)$, $X_2(t)$, and $X_3(t)$ Remain Balanced}}{5}} \bibstyle{chicago} \bibdata{DBP} \bibcite{AchlioptasMolloy}{\citeauthoryear {Achlioptas and Molloy}{Achlioptas and Molloy}{1997}} \bibcite{KnuthStableMariagesBook}{\citeauthoryear {Knuth}{Knuth}{1976}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{15.0pt} \newlabel{tocindent1}{21.0pt} \newlabel{tocindent2}{30.0pt} \newlabel{tocindent3}{0pt} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.3}{$\@mathbb {P}_tZ_{t+1}^2$ is Bounded for by a Constant}}{6}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References\begingroup \@temptokena {{\uppercase {References}}{\uppercase {References}}}\xdef {\uppercase {References}}{\uppercase {References}}{}\mark {}\endgroup }}{6}}