diff --git a/papers/even_level_graph_generators/paper.aux b/papers/even_level_graph_generators/paper.aux index 03be82f..d8e08dc 100644 --- a/papers/even_level_graph_generators/paper.aux +++ b/papers/even_level_graph_generators/paper.aux @@ -17,40 +17,40 @@ \providecommand\HyField@AuxAddToFields[1]{} \providecommand\HyField@AuxAddToCoFields[2]{} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{section.1}\protected@file@percent } -\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Definitions}}{1}{section.2}\protected@file@percent } -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces BFS levels from the degree-$3$ vertex source $S = \{4\}$. The source is level $0$, its three neighbours are level $1$, and the remaining vertices are level $2$. Colour encodes the level.}}{1}{figure.1}\protected@file@percent } -\newlabel{fig:levels}{{1}{1}{BFS levels from the degree-$3$ vertex source $S = \{4\}$. 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Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge.}}{7}{figure.5}\protected@file@percent } +\newlabel{fig:n21-duals}{{5}{7}{The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. 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The first family is the +\emph{bridge-derived level graphs}: starting from an \emph{Even Level +Graph} -- a triangulation all of whose level cycles, measured by +breadth-first distance from a chosen source vertex, are even -- we apply +\emph{bridge switches}, edge switches that never close a cycle in either +parity subgraph. Every such graph is $4$-colorable, inheriting the parity +$2$-coloring of an Even Level Graph. The second family is the +\emph{intertwining trees}: triangulations whose vertices split into two +sets each inducing a tree, which are $4$-colorable by coloring the two +trees from disjoint pairs of colors. We conjecture that every maximal +planar graph is a bridge-derived level graph, an intertwining tree, or +both; since both families are $4$-colorable by construction, the +conjecture would give a constructive proof of the four color theorem for +triangulations, and hence for all planar graphs. We show that a +triangulation is an intertwining tree exactly when its dual is +Hamiltonian, so every triangulation on at most $20$ vertices is an +intertwining tree and the first possible failures occur at $n = 21$, at +the six duals of the Holton--McKay graphs. We verify that all six are +bridge-derived level graphs, confirming the conjecture in its first +nontrivial case. \end{abstract} \maketitle \section{Introduction} +The four color theorem states that every planar graph is properly +$4$-colorable. It suffices to prove this for \emph{maximal} planar graphs +(triangulations), since every planar graph is a spanning subgraph of one. +The known proofs proceed by reducing a hypothetical minimal counterexample, +either through computer-checked unavoidable configurations or through +discharging. + +We take a constructive view. Instead of coloring an arbitrary +triangulation, we ask which triangulations can be \emph{assembled} by +operations that manifestly preserve $4$-colorability, and whether those +operations reach all of them. We study two such constructions, and our +motivating question is whether the two together are sufficient: does every +maximal planar graph on $n$ vertices arise from one of them? + +The first construction builds on the \emph{level} structure of a +triangulation. Fixing a source vertex and taking breadth-first levels, an +\emph{Even Level Graph} (Definition~\ref{def:even-level-graph}) is a +triangulation whose level cycles are all even; equivalently both of its +parity subgraphs are bipartite, and a $2$-coloring of each parity subgraph +-- two colors for the even-level vertices, two for the odd -- is a proper +$4$-coloring (Theorem~\ref{thm:even-level-4colorable}). From an Even Level +Graph we generate further triangulations by edge switches; restricting to +\emph{bridge switches} (Definition~\ref{def:bridge-switch}), which add an +edge to a parity subgraph only when it is a bridge there, guarantees that +no new cycle -- and in particular no odd cycle -- ever appears in a parity +subgraph. The resulting \emph{bridge-derived level graphs} +(Definition~\ref{def:bridge-derived-level-graph}) therefore remain +$4$-colorable by the same parity coloring. + +The second construction is purely combinatorial: an \emph{intertwining +tree} (Definition~\ref{def:intertwining-tree}) is a triangulation whose +vertex set partitions into two parts each inducing a tree. Coloring one +tree from $\{1,2\}$ and the other from $\{3,4\}$ is a proper $4$-coloring, +since edges inside a part join differently-colored tree vertices and edges +across the parts join the disjoint color sets. + +Our central question is whether these two families exhaust all +triangulations +(Conjecture~\ref{conj:every-triangulation-derived}). As both families +consist of $4$-colorable graphs, an affirmative answer would constitute a +constructive proof of the four color theorem for triangulations. + +We connect the two constructions through duality: a triangulation is an +intertwining tree if and only if its dual is Hamiltonian +(Theorem~\ref{thm:intertwining-iff-hamiltonian-dual}). Tait's conjecture -- +that every $3$-connected cubic planar graph is Hamiltonian -- fails first +at $38$ vertices, where Holton and McKay found exactly six counterexamples; +dually, every triangulation on at most $20$ vertices is an intertwining +tree, and the first triangulations that are not are the six $21$-vertex +duals of the Holton--McKay graphs. These six are therefore the first +nontrivial test of the conjecture, and we verify that all six are +bridge-derived level graphs -- each at most four bridge switches from an +Even Level Graph, and two of them Even Level Graphs already. + \section{Definitions} Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation)