even_level: title/abstract/intro -- frame conjecture as stronger than the 4CT
Retitle to "Even Level Graph Generators: a constructive conjecture
stronger than the Four Color Theorem" and state explicitly in the
abstract and introduction that the conjecture implies the four color
theorem but is strictly stronger: a 4-coloring grouped {1,2}|{3,4} is
exactly a partition into two bipartite-inducing parts, so 4CT is the bare
existence of such a partition, whereas the conjecture demands it be
realized constructively (bridge-switch level parity, or two induced
trees). Hence a proof is a new constructive proof of 4CT, and the
conjecture is at least as hard -- very likely harder.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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\newlabel{fig:edge-switch}{{3}{4}{An edge switch on the level cycle of Figure~\ref {fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes $1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex colours indicate the original levels in $G$}{figure.3}{}}
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\newlabel{fig:parity-subgraph}{{4}{4}{Parity subgraphs of $G' = T$ with respect to the level structure of Figure~\ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \bmod 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. Right: the odd parity subgraph $O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows that $O_{G,S}(G')$ is not bipartite for this choice of $G'$}{figure.4}{}}
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\title{Even Level Graph Generators}
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\title[Even Level Graph Generators]{Even Level Graph Generators:\\
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a constructive conjecture stronger than the Four Color Theorem}
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||||
@@ -93,9 +94,14 @@ $2$-coloring of an Even Level Graph. The second family is the
|
||||
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
|
||||
both. Since both families are $4$-colorable by construction, the
|
||||
conjecture implies the four color theorem for triangulations, and hence
|
||||
for all planar graphs; in fact it is \emph{strictly stronger}, demanding
|
||||
not merely that a $4$-coloring exist but that every triangulation be
|
||||
assembled by one of these two explicit constructions. A proof would
|
||||
therefore be a new, constructive proof of the four color theorem -- and
|
||||
correspondingly the conjecture is at least as hard, and very likely
|
||||
harder, than that theorem. 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
|
||||
@@ -153,7 +159,25 @@ 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.
|
||||
constructive proof of the four color theorem for triangulations, and
|
||||
hence for all planar graphs.
|
||||
|
||||
We emphasize that the conjecture is a \emph{stronger} statement than the
|
||||
four color theorem, not an equivalent reformulation of it. A proper
|
||||
$4$-coloring with its colors grouped as $\{1,2\}\mid\{3,4\}$ is exactly a
|
||||
partition of the vertices into two parts each inducing a bipartite
|
||||
subgraph, so the four color theorem is precisely the assertion that every
|
||||
triangulation admits such a partition. The conjecture asserts strictly
|
||||
more: that the partition can be realized \emph{constructively} -- as the
|
||||
level parity of an Even Level Graph reached by bridge switches, or as a
|
||||
split into two induced \emph{trees}. The four color theorem alone supplies
|
||||
neither construction; bridge-derivability in particular is a reachability
|
||||
condition well beyond the bare existence of a $4$-coloring, so the
|
||||
conjecture implies the four color theorem but is not implied by it.
|
||||
A proof would accordingly be a new, constructive proof of the four color
|
||||
theorem, and the conjecture is at least as hard to settle -- and, absent
|
||||
any structural characterization of the bridge-derived family, very likely
|
||||
harder.
|
||||
|
||||
We connect the two constructions through duality: a triangulation is an
|
||||
intertwining tree if and only if its dual is Hamiltonian
|
||||
|
||||
Reference in New Issue
Block a user