deletions | additions       diff --git a/section_Colorability_and_unanimity_In__.tex b/section_Colorability_and_unanimity_In__.tex  index 23d8e96..57aace4 100644  --- a/section_Colorability_and_unanimity_In__.tex  +++ b/section_Colorability_and_unanimity_In__.tex 
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The possibility of this extreme distribution of opinions relies on the graph $G_{p_j}$ being $2$-colorable, which is equivalent to $G_{p_j}$ not containing any cycle of odd length. However, non $2$-colorability is not a sufficient condition for convergence of UPs in general: a simple cycle of three agents, for instance, is not $2$-colorable but does not guarantee convergence (as illustrated above for BDPs).   Nevertheless, there is a class of influence profiles for which being $2$-colorable is a necessary condition of non-convergence of UPs, the \emph{symmetric} influence profiles:  \begin{proposition} \begin{theorem}  Let $\G=(G_{p_1},\dots,G_{p_m})$ be a symmetric (and non-necessarily functional) influence profile and $\O=(O_1,\dots,O_n)$ be an opinion profile.  The UP converges for $\O$ on $\G$ iff:  \begin{itemize}  \item[] For all $j\in\{1,\dots,m\}$, $G_pj$ is not $2$-colorable, or  \item[] There is a $j\in\{1,\dots,m\}$, such that $\O$ properly colors $G_pj$: that:  for all $i,j\in\$ such that $i,j\in\N$ if  $j\in R_j(i)$, $O_i{p_j}\neq O_j(p_j)$. \end{itemize}  \end{proposition} \end{theorem}  \subsection{modal logic and colorability}  \subsection{backdrop on stability and colorability}  %%%%%%%%%%%%%%%%%%%%%%%%BELOW THIS POINT IS ONLY OLDER STUFF%%%%%%%%%%%%%%  There are two stability-related questions, given an initial model: