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\title{Cut-and-depth-label: a procedure for labelling half-graphs of\\
       a $6$-edge cut by ``distance to the cut''}
\author{}
\date{}

\newtheorem*{obs}{Observation}
\newtheorem*{prop}{Proposition}
\newtheorem*{definition}{Definition}

\begin{document}
\maketitle

\section*{Procedure}

Given a maximal planar graph $G$ and its dual $G'$:

\begin{enumerate}
  \item Find a $6$-edge cut $C \subseteq E(G')$ partitioning
        $V(G')$ into $S$ and $V \setminus S$ with both sides
        non-empty.
  \item Remove the cut edges to obtain two graphs
        $G'_0 := G'[S] \cup \emptyset_C$ and
        $G'_1 := G'[V \setminus S] \cup \emptyset_C$ (each a cubic
        graph minus boundary edges).
  \item In each $G'_i$:
        \begin{enumerate}
          \item Let $V_i$ be the set of vertices of degree $2$ in
                $G'_i$ (i.e.\ original cubic vertices incident to
                exactly one cut edge; the cut edge accounts for the
                missing edge, so the vertex sits at degree $3 - 1 =
                2$).
          \item For each $v \in V_i$, attach a new pendant edge to
                a fresh vertex, and \textbf{label these pendant
                edges with depth $0$}.
          \item For $d = 0, 1, 2, \ldots$: label every unlabelled
                edge that shares a vertex with a depth-$d$ edge with
                depth $d + 1$.
          \item Stop when every edge has a depth label.
        \end{enumerate}
\end{enumerate}

\noindent\textbf{Interpretation.}  The depth labels give a BFS
distance in the line graph of $G'_i$ starting from the pendants
added at the cut.  Equivalently, the depth of an edge $e$ in $G'_i$
is the minimum number of edges traversed (via shared-vertex
adjacency) to reach a pendant.

\noindent\textbf{Caveat.}  If the cut $C$ is not a \emph{matching}
cut (i.e., the $6$ cut edges share vertices), then some boundary
vertices have degree $< 2$ in $G'_i$ and do not receive a pendant
under the strict reading of step~3(a).  When the cut is a
matching, each of the $12$ boundary vertices ($6$ per side) has
degree exactly $2$ and receives a pendant; the construction is
symmetric.

\section*{Example: Holton-McKay graph \#0}

We apply the procedure to the first of the six non-Hamiltonian
$38$-vertex cubic plane graphs found by Holton and McKay (loaded
from \texttt{papers/even\_level\_graph\_generators/experiments/nonham38m4.pc}).
This $G'$ is itself a cubic plane graph; its dual $G$ is a
$21$-vertex triangulation.

\paragraph{The chosen 6-edge cut.}
Greedy search over $128$ distinct $6$-edge cuts in $G'$, preferring
\emph{matching cuts} (both sides have $6$ distinct boundary
vertices) and then balanced $|S|$, returns
\[
   |S| = 10, \qquad
   C = \{(34, 29),\, (35, 30),\, (26, 22),\, (27, 23),\,
        (28, 24),\, (31, 25)\}.
\]
This is a matching cut: the $6$ edges have $12$ distinct endpoints,
$6$ on each side.

\paragraph{Resulting half-graphs.}

\begin{center}
\begin{tabular}{lcc}
\toprule
& $G'_0$ & $G'_1$ \\
\midrule
$|S|$ & $10$ & $28$ \\
Original vertices in $G'_i$ & $10$ & $28$ \\
$|V_i|$ (pendants added) & $6$ & $6$ \\
Total vertices in $G'_i$ & $16$ & $34$ \\
Total edges & $18$ & $45$ \\
Max depth assigned & $2$ & $7$ \\
\bottomrule
\end{tabular}
\end{center}

\noindent
Multi-cut vertices (degree $< 2$ in induced subgraph): \emph{none}
on either side, since the cut is a matching.

\paragraph{Visualization.}

\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{fig_cut_depth_label.png}
\caption{Cut-and-depth-label procedure on Holton-McKay graph \#0
($38$ vertices, $57$ edges, cubic planar non-Hamiltonian).  The
$6$-edge matching cut splits $G'$ into a $10$-vertex piece $G'_0$
(left) and a $28$-vertex piece $G'_1$ (right).  Pendant edges
(dashed, dark purple) are at depth $0$; the colour gradient
encodes depth $0$ through max depth via BFS in the line-graph
sense.  Orange squares are the new pendant vertices; red circles
are the boundary vertices in $V_i$; gray circles are interior
vertices.}
\label{fig:cut-depth-label}
\end{figure}

\section*{Cut tires}

The depth labelling on $G'_i$ organises its edges into layers indexed
by distance to the cut.  Each layer gives rise to a family of
``cut tires'' that play the same structural role as the tires of
\texttt{paper.tex} but are derived from the cut rather than from a
level source in the primal $G$.

\begin{definition}[Cut tire]
Let $G'_i$ be a cut half (Step~3 above) with edge depth labelling
$\mathrm{depth} : E(G'_i) \to \mathbb{Z}_{\ge 0}$.  Fix $d > 0$ and
let $H_d \subseteq G'_i$ be the subgraph induced on the edges of
depth $d$ (vertex set $=$ endpoints of depth-$d$ edges, edges $=$
depth-$d$ edges).  Equip $H_d$ with the planar embedding inherited
from $G'_i$.

For each face $f$ of $H_d$, the \emph{cut tire at $(d, f)$} is the
graph built as follows:
\begin{itemize}
  \item Start with the boundary walk of $f$ as the \emph{face
        boundary} (every edge here is of depth $d$ in $H_d$).
  \item For each vertex $v$ on the boundary walk of $f$ that has
        degree exactly $2$ in $H_d$:
        \begin{itemize}
          \item If $v$ is incident in $G'_i$ to an edge of depth
                $d - 1$, add a fresh vertex $n_v$ and a fresh edge
                $\{v, n_v\}$, labelled an \emph{out spoke}.
          \item If $v$ is incident in $G'_i$ to an edge of depth
                $d + 1$, add a fresh vertex $n_v$ and a fresh edge
                $\{v, n_v\}$, labelled an \emph{in spoke}.
        \end{itemize}
\end{itemize}

\noindent\emph{Convention.}  ``Out'' means toward lower depth (toward
the cut, the outer boundary of $G'_i$); ``in'' means toward higher
depth (deeper into $G'_i$).  Because $G'_i$ is cubic at every
original vertex, a degree-$2$ boundary vertex $v$ has exactly one
non-$H_d$ edge in $G'_i$, so $v$ contributes exactly one spoke (either
in or out, not both).

\noindent\emph{Structural remark.}  Under this definition, each cut
tire is intrinsically a graph of the form ``cycle (or closed walk) $+$
pendants attached at simple boundary vertices,'' structurally
isomorphic to the partial tire dual $D(T)$ of \texttt{paper.tex}
(Proposition 1.8: $D(T) \cong C_{n+m} \circ K_1$ in the spoke-only
case).  The labelled in/out spokes are the analogue of $D(T)$'s
leaves; the face boundary plays the role of $T'_{\mathrm{ann}}$.
\end{definition}

\paragraph{Relation to the existing tire framework.}  Under the
correspondence between primal level structure (\texttt{paper.tex})
and dual depth labelling, a cut tire on $G'_i$ at $(d, f)$ is
isomorphic as a graph to the partial tire dual $D(T)$ of a tire
$T$ in \texttt{paper.tex} (Def.~1.7):
\begin{itemize}
  \item face boundary at depth $d$ $\longleftrightarrow$
        $T'_{\mathrm{ann}}$, the cycle (or closed walk) of annular
        dual vertices;
  \item in/out spokes $\longleftrightarrow$ the leaves of $D(T)$
        (Def.~1.7's degree-$1$ pendants attached to the cycle);
  \item the cut tire's planar embedding $\longleftrightarrow$
        $D(T)$'s embedding.
\end{itemize}
The cut tire is therefore not just analogous to but \emph{is} a
partial tire dual (up to relabelling), with depth-from-cut playing
the role of level depth from the primal source.  Consequently every
proposition about $D(T)$ in paper.tex --- chromatic polynomial
counts, $S_3$-orbit structure, rainbow conjecture, etc.\ --- applies
verbatim to each cut tire.

\paragraph{Example on $G'_1$ (Holton-McKay \#0, $V \setminus S$ half).}

\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{fig_cut_tire.png}
\caption{Cut tires on $G'_1$ at depths $d = 1, 2, 4, 5, 6$.  In each
panel, blue solid edges form the face boundary at depth $d$; red
dashed edges are inner spokes (depth $d - 1$); green dashed edges
are outer spokes (depth $d + 1$).  Vertices on the face boundary are
highlighted; the rest of $G'_1$ is faded.}
\label{fig:cut-tire}
\end{figure}

In this example, counting only degree-$2$ boundary vertices (which
under the redefinition each contribute exactly one in/out spoke):
\begin{itemize}
  \item $d = 1$: face length $12$, $5$ out spokes (toward depth-$0$
        pendants) $+\ 0$ in spokes.  The outermost cut tire,
        immediately adjacent to the cut.  No in spokes because every
        depth-$2$ neighbour of a face-boundary vertex is at a
        higher-degree vertex of $H_1$.
  \item $d = 2$: face length $7$ (one of two symmetric faces in
        $H_2$), $4$ out $+\ 3$ in spokes.
  \item $d = 4$: face length $8$, $2$ out $+\ 1$ in.
  \item $d = 5$: face length $14$, $4$ out $+\ 2$ in.
  \item $d = 6$: face length $12$, $3$ out $+\ 2$ in.  Innermost cut
        tire in this chain.
\end{itemize}

\section*{A looser chain pigeonhole hypothesis}

The rainbow conjecture and K\"onig-lift conjecture (from
\texttt{rainbow\_proof.tex} and
\texttt{worst\_case\_proof\_sketch.tex}) are statements about
specific structured face boundaries (antipodal-chord SP).  Generic
cut tires in our example do not have those specific structures, so
those conjectures don't apply directly.  Empirically
(\texttt{cut\_tire\_conjecture\_tests.tex}), cut-tire projections
are nonetheless large and well-behaved:

\begin{itemize}
  \item Every cut tire's joint projection $\pi(T) :=
        \{(\sigma_{\mathrm{out}}, \sigma_{\mathrm{in}}) : \chi
        \text{ proper}\}$ is $S_3$-closed (universal, follows from
        color symmetry of proper edge $3$-colouring).
  \item All $S_3$-orbits in $\pi(T)$ have size exactly $3$ (the
        constant-colour orbit, when present) or $6$ (the generic
        orbit using all $3$ colours); no size-$2$ stabilizer orbits
        occur.
  \item Non-trivial cut tires have $|\pi(T)| \gg 6$: e.g.\ at
        depth~$2$ the projection has size $126 = 21 \cdot 6$
        ($21$ full $S_3$-orbits), and at depth~$6$ size $93 = 1
        \cdot 3 + 15 \cdot 6$.
\end{itemize}

This motivates a much weaker conjecture than the rainbow/K\"onig
ones --- one that does not require special face-boundary structure
and may be easier to prove:

\begin{quote}
\textbf{Loose chain pigeonhole conjecture (cut tires, $k \ge 2$
form).}  Let $T = T_d^{(i, f)}$ be a cut tire with total spoke
count $|\text{in spokes}| + |\text{out spokes}| \ge 2$.  Then the
joint projection $\pi(T)$ is non-empty, $S_3$-closed, and contains
at least one full $S_3$-orbit of size $6$ (i.e.\ a $\sigma$ using
$\ge 2$ distinct colours), giving $|\pi(T)| \geq 6$.

Consequently, for the chain $T_1^{(i)} \to T_2^{(i)} \to \cdots$ on
each side of a cut, the realisable cut-configuration set
$\mathcal{R}_i$ contains a full $S_3$-orbit, and the intersection
$\mathcal{R}_0 \cap \mathcal{R}_1$ contains a common $S_3$-orbit
(of size $6$), forcing $\mathcal{R}_0 \cap \mathcal{R}_1 \ne
\emptyset$.
\end{quote>

\noindent\emph{Restriction rationale.}  Cut tires with exactly $k =
1$ in/out spoke have $\pi(T) \subseteq \{1, 2, 3\}$, where every
single-colour $\sigma$ has $S_3$-stabilizer of order $2$, so its
$S_3$-orbit has size $6 / 2 = 3$, never $6$.  Such tires trivially
violate ``$|\pi(T)| \ge 6$''
(see \texttt{loose\_conjecture\_counterexamples.tex}, where two
$k = 1$ cut tires are exhibited in $G'_1$ of Holton-McKay \#0).
The $k \ge 2$ restriction excludes these trivially excluded cases.

\paragraph{Justification of the per-tire half.}  The per-tire half
is essentially Proposition~1.13 of \texttt{paper.tex}: a spoke-only
cut tire (face boundary a simple cycle of length $n$) has $2^n +
2(-1)^n$ proper edge $3$-colourings, which is $\ge 6$ whenever
$n \ge 3$.  By $S_3$-closure, the orbit count is at least $1$,
hence $\ge 6$ distinct projections (modulo the size-$3$ constant
orbit edge case).  For cut tires with non-simple face boundary
(e.g.\ $\theta(1, p, q)$-shape) similar lower bounds follow from
the chromatic polynomial of the corresponding $D(T)$ structure (see
the menagerie note for explicit closed forms when chord matchings
are present).

\paragraph{Justification of the chain half.}  Chain composition
preserves $S_3$-symmetry (each step is $S_3$-equivariant), so
$\mathcal{R}_i$ inherits $S_3$-closure from the per-tire projection.
The non-trivial claim is that the orbit structure \emph{survives}
composition: an $S_3$-orbit in $\pi(T_d)$ projects to some
non-empty $S_3$-closed subset on the out-spoke side, and this
propagates layer-by-layer.

\paragraph{What this would (and would not) prove.}  If the loose
conjecture holds, chain pigeonhole at the cut gives $\mathcal{R}_0
\cap \mathcal{R}_1 \neq \emptyset$, which (under the minimum-
counterexample reduction) contradicts $G'$ being a 4CT
counterexample.  This is the same final conclusion as the
rainbow/K\"onig route, with weaker hypotheses but stronger
empirical support.

It would \emph{not} prove the rainbow conjecture itself (which is a
strict structural claim about the antipodal-chord SP case), nor
the K\"onig-lift conjecture's specific construction; it would only
recover the bottom-line chain pigeonhole result.

\paragraph{Status.}  Strong empirical support:
\begin{itemize}
  \item Broader sweep across all $6$ Holton-McKay graphs, $3$ cuts
        per graph, both sides
        (\texttt{experiments/loose\_conjecture\_sweep.py};
        data: \texttt{loose\_conjecture\_sweep\_data.txt}):
        \begin{itemize}
          \item $287$ total cut tires examined.
          \item $154$ with $k \ge 2$ in/out spokes.
          \item $107$ of these had $\le 14$ edges and were
                brute-force verified; the remaining $47$ were too
                large to enumerate exhaustively.
          \item \textbf{All $107$ passed: $|\pi(T)| \ge 6$ with a
                full $S_3$-orbit.}
          \item \textbf{$0$ counterexamples found.}
        \end{itemize}
  \item The per-tire half (existence of a multi-color spoke
        configuration) is provable for spoke-only cut tires (face
        boundary a simple cycle of length $n \ge 3$) via Prop~1.13:
        $2^n + 2(-1)^n$ total proper edge $3$-colourings, of which
        at most $6$ are ``all-spokes-same-color''
        (only possible when $n$ is even and the cycle admits a
        $2$-edge-colouring), leaving $\ge 2^n + 2(-1)^n - 6 > 0$
        multi-color configurations for $n \ge 3$ even or $n \ge 3$
        odd.
  \item The chain half is the genuinely open question --- it requires
        showing that the per-tire $S_3$-orbit structure composes
        coherently through the depth chain.
\end{itemize}

\section*{Connection to chain pigeonhole / 4CT reducibility}

The procedure mirrors the $4$CT cut-and-reglue scheme
(\texttt{rainbow\_proof.tex}, \texttt{worst\_case\_proof\_sketch.tex},
\texttt{two\_approaches\_comparison.tex}) at the structural level.
After cutting, each $G'_i$ is a cubic-minus-boundary graph; the
pendant additions formally restore cubicity at the $6$ degree-$2$
boundary vertices.  The depth label on each edge measures its
``distance to the cut.''

For a minimum counterexample to the 4CT (i.e., a cubic plane $G'$
with no proper $3$-edge-colouring), the depth labels organise each
$G'_i$ into concentric layers indexed by distance to the cut.  The
$3$-edge-colourings of $G'_i$ must extend a colouring at the
depth-$0$ pendants (= a ring colouring at the cut); the BFS
ordering by depth is the natural induction order for propagating
the colouring inward.

In the tire framework, the cut cycle $\gamma$ in the primal $G$
(corresponding to the $6$-edge cut in $G'$ via planar duality) plays
the role of the tire's inner boundary on one side and outer boundary
on the other.  The depth label on $G'_i$-edges is exactly the dual
analogue of plane depth from $\gamma$ (cf.\ the level-cycle
generalization discussion in the recent conversation).

\section*{Limitations of this example}

\begin{itemize}
  \item Holton-McKay graphs are cyclically $5$-edge-connected (not
        $6$-edge-connected), so $6$-edge cuts are not the minimum
        cyclic cut.  The smallest cyclic edge cuts in this graph are
        size $5$.
  \item The matching $6$-cut found is highly imbalanced ($|S|=10$
        vs.\ $|S^c|=28$).  Searching among the $128$ distinct
        $6$-edge cuts for a balanced matching cut may give better
        examples.
  \item Depth labels propagate via the line graph, not via vertex
        BFS.  An alternative procedure would label \emph{vertices}
        by BFS distance from the boundary; both yield similar
        layered structures but with slightly different counts.
\end{itemize}

\end{document}
