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>

papers/even_level_graph_generators/paper.tex

@@ -59,7 +59,8 @@
 
 \begin{document}
 
-\title{Even Level Graph Generators}
+\title[Even Level Graph Generators]{Even Level Graph Generators:\\
+a constructive conjecture stronger than the Four Color Theorem}
 
 %    Remove any unused author tags.
 
@@ -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