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PDF statistics: - 102 PDF objects out of 1000 (max. 8388607) - 61 compressed objects within 1 object stream + 110 PDF objects out of 1000 (max. 8388607) + 66 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 1 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/paper.pdf b/paper.pdf index ad3733d..91b6acb 100644 Binary files a/paper.pdf and b/paper.pdf differ diff --git a/paper.synctex.gz b/paper.synctex.gz index b13b335..f6124ca 100644 Binary files a/paper.synctex.gz and b/paper.synctex.gz differ diff --git a/paper.tex b/paper.tex index fb370ec..f1cffd5 100644 --- a/paper.tex +++ b/paper.tex @@ -258,6 +258,44 @@ Let $\psi \in \Phi_4(G'_{ij})$, so $|\psi(N_{ij}(v_0))| = 4$. By definition of t The first equality expands $N(v_0)$; the second uses $\widetilde{\psi}(a_i) = \widetilde{\psi}(a_j) = \psi(a_{ij})$ and $\widetilde{\psi}(u) = \psi(u)$ for $u \notin \{a_i, a_j\}$; the third uses $N_{ij}(v_0) = \{a_{ij}\} \cup (N(v_0) \setminus \{a_i,a_j\})$. Therefore $|\widetilde{\psi}(N(v_0))| = |\psi(N_{ij}(v_0))| = 4$. Since also $\widetilde{\psi} \in \Phi$ by the previous lemma, we conclude $\widetilde{\psi} \in \Phi_4$. \end{proof} +\subsection{Locked Colorings} + +\begin{definition} +Let $a_i, a_j \in N(v_0)$ be non-adjacent in $G'$ (where $G' = G - \{v_0, v_1\}$ as in Section 2.1). A coloring $\phi \in \Phi$ is \emph{locked relative to $\{a_i, a_j\}$} if +\[ + |\phi(N_{G'}(a_i))| > 2 \quad \text{or} \quad |\phi(N_{G'}(a_j))| > 2. +\] +Denote the set of all such colorings by $\Lambda_{ij} \subseteq \Phi$. +\end{definition} + +Intuitively, a coloring is locked relative to $\{a_i, a_j\}$ when the neighborhood of at least one of the two vertices is colored with enough distinct colors to obstruct a Kempe chain swap that would free a color for $v_0$. + +\begin{lemma} +Let $G'_{ij} \in \mathcal{M}$ and let $\psi \in \Phi(G'_{ij})$. If $|\psi(N_{G'_{ij}}(v_1))| \leq 3$, then $\psi$ is the induced coloring of some locked coloring in $\Phi$. +\end{lemma} + +\begin{proof} +The extension $\widetilde{\psi} \in \Phi$ (by the lemma in Section 3.1). Suppose for contradiction that $\widetilde{\psi} \notin \Lambda_{ij}$, i.e., $\widetilde{\psi}$ is not locked relative to $\{a_i, a_j\}$. Then +\[ + |\widetilde{\psi}(N_{G'}(a_i))| \leq 2 \quad \text{and} \quad |\widetilde{\psi}(N_{G'}(a_j))| \leq 2. +\] +Since $\widetilde{\psi}$ agrees with $\psi$ on all vertices of $G'_{ij}$, the neighborhoods of $a_i$ and $a_j$ in $G'$ each use at most 2 colors under $\widetilde{\psi}$. With so few colors in each neighborhood, a Kempe chain swap can be performed in $\widetilde{\psi}$ to give $a_i$ and $a_j$ the same color, freeing a fourth color for $v_0$. Simultaneously, since $|\psi(N_{G'_{ij}}(v_1))| \leq 3$, the vertex $v_1$ can be assigned the remaining color not used by its neighbors. Together these assignments extend $\widetilde{\psi}$ to a proper 4-coloring of all of $G$, contradicting the minimality of $G$ as a counterexample. Therefore $\widetilde{\psi} \in \Lambda_{ij}$, and $\psi$ is the induced coloring of the locked coloring $\widetilde{\psi}$. +\end{proof} + +\begin{lemma} + Now prove that all colorings of every merged graph relative to $\{a_i, a_j\}$ must be a locked coloring. +\end{lemma} + +\begin{proof} +\end{proof} + +\begin{lemma} +If all $\phi \in \Phi$ are locked colorings with respect to all pairs of non adjacent vertices $\{a_i, a_j\} \in N(v_0)$, then all colorings of all merged graphs with respect to ${a_k, a_l} \in N(v_1)$ require 4 colors for $N_(v_0)$. +\end{lemma} + +\begin{proof} +\end{proof} + \end{document} %-----------------------------------------------------------------------