Keep only the resulting-duals figure at n=21
Remove the witness-ELG figure (former Fig. 5); keep the six resulting duals with their introduced green bridge edges. Fix the dangling cross-reference in the caption. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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@@ -426,29 +426,16 @@ witnesses are step-verified.}
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\label{tab:n21}
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\label{tab:n21}
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\end{table}
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\end{table}
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\begin{figure}[ht]
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\centering
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\includegraphics[width=\textwidth]{figures/n21_elgs.png}
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\caption{The witness Even Level Graph for each of the six Holton--McKay
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duals, drawn as a crossing-free planar graph and coloured by parity (blue
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even, orange odd, with respect to the fixed level-parity labelling). The
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dashed red edges are the same-parity edges that the bridge switches flip;
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flipping them yields the corresponding dual in
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Figure~\ref{fig:n21-duals}. Duals $1$ and $2$ are Even Level Graphs
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outright, so no edge is flipped.}
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\label{fig:n21-elgs}
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\end{figure}
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\begin{figure}[ht]
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\begin{figure}[ht]
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\centering
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\centering
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\includegraphics[width=\textwidth]{figures/n21_duals.png}
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\includegraphics[width=\textwidth]{figures/n21_duals.png}
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\caption{The six Holton--McKay duals, drawn as crossing-free planar graphs
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\caption{The six Holton--McKay duals, drawn as crossing-free planar graphs
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with the same parity colouring. The solid green edges are the bridge edges
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and coloured by parity (blue even, orange odd, with respect to the fixed
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introduced by the switches from the Even Level Graphs of
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level-parity labelling). The solid green edges are the bridge edges
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Figure~\ref{fig:n21-elgs}. Each green edge is a bridge of its parity
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introduced by the bridge switches from each dual's witness Even Level
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subgraph, so no new cycle -- and in particular no odd cycle -- is created;
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Graph. Each green edge is a bridge of its parity subgraph, so no new cycle
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duals $1$ and $2$ coincide with their Even Level Graphs and have no added
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-- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide
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edge.}
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with their Even Level Graphs and have no added edge.}
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\label{fig:n21-duals}
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\label{fig:n21-duals}
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\end{figure}
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\end{figure}
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