From ea6c99ec6dd18bdba7928de06fe539b35d469f1f Mon Sep 17 00:00:00 2001 From: charlesen <296161+charlesen@users.noreply.github.com> Date: Sat, 1 Aug 2026 14:00:40 +0000 Subject: [PATCH] =?UTF-8?q?Add=20structural=20analysis=20and=20proof=20str?= =?UTF-8?q?ategy=20for=20Erd=C5=91s-Gy=C3=A1rf=C3=A1s=20conjecture?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit This commit introduces a detailed mathematical exploration of the Erdős-Gyárfás conjecture (assigned problem number 31). The changes include: - Generating English and French LaTeX files detailing axiomatic definitions, literature context, lemmata, proof strategies, and a Lean 4 formalization architecture. - Compiling these LaTeX files into PDFs ready for review. - Creating corresponding README and README.fr.md files for documentation. - The files are safely placed within the `inprogress/31-Erdos-Gyarfas` directory as the problem remains a partial open solution. Co-authored-by: google-labs-jules[bot] <161369871+google-labs-jules[bot]@users.noreply.github.com> --- .../31-Erdos-Gyarfas/31-Erdos-Gyarfas-FR.pdf | Bin 0 -> 193998 bytes .../31-Erdos-Gyarfas/31-Erdos-Gyarfas-FR.tex | 90 ++++++++++++++++++ .../31-Erdos-Gyarfas/31-Erdos-Gyarfas.pdf | Bin 0 -> 194380 bytes .../31-Erdos-Gyarfas/31-Erdos-Gyarfas.tex | 89 +++++++++++++++++ inprogress/31-Erdos-Gyarfas/README.fr.md | 7 ++ inprogress/31-Erdos-Gyarfas/README.md | 7 ++ 6 files changed, 193 insertions(+) create mode 100644 inprogress/31-Erdos-Gyarfas/31-Erdos-Gyarfas-FR.pdf create mode 100644 inprogress/31-Erdos-Gyarfas/31-Erdos-Gyarfas-FR.tex create mode 100644 inprogress/31-Erdos-Gyarfas/31-Erdos-Gyarfas.pdf create mode 100644 inprogress/31-Erdos-Gyarfas/31-Erdos-Gyarfas.tex create mode 100644 inprogress/31-Erdos-Gyarfas/README.fr.md create mode 100644 inprogress/31-Erdos-Gyarfas/README.md diff --git a/inprogress/31-Erdos-Gyarfas/31-Erdos-Gyarfas-FR.pdf 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+\usepackage{amsmath, amssymb, amsthm} +\usepackage{geometry} +\geometry{margin=1in} + +\title{Analyse Structurale et Stratégie de Preuve Constructive de la Conjecture d'Erd\H{o}s-Gy\'{a}rf\'{a}s} +\author{Charles EDOU NZE\thanks{Charles EDOU NZE, chercheur indépendant}} +\date{} + +\newtheorem{theorem}{Théorème}[section] +\newtheorem{lemma}[theorem]{Lemme} +\newtheorem{definition}[theorem]{Définition} +\newtheorem{conjecture}[theorem]{Conjecture} + +\begin{document} +\maketitle + +\begin{abstract} +Ce document présente une formulation axiomatique, une revue de la littérature et une architecture de preuve structurée pour la conjecture d'Erd\H{o}s-Gy\'{a}rf\'{a}s en théorie des graphes. La conjecture stipule que tout graphe dont le degré minimum est d'au moins 3 contient un cycle dont la longueur est une puissance de 2. Nous décomposons le problème en plusieurs lemmes intermédiaires et fournissons une architecture adaptée à la vérification formelle dans Lean 4. +\end{abstract} + +\section{Introduction et Définitions Axiomatiques} +La conjecture d'Erd\H{o}s-Gy\'{a}rf\'{a}s est un problème ouvert fondamental en théorie des graphes extrémaux. Soit $G = (V, E)$ un graphe simple, fini et non orienté. Soit $\delta(G)$ le degré minimum des sommets dans $G$. + +\begin{definition}[Graphe et Degré Minimum] +Soit $V$ un ensemble fini et $E \subseteq \binom{V}{2}$ l'ensemble des arêtes. Le degré minimum est défini par : +\[ +\delta(G) = \min_{v \in V} \deg(v) +\] +où $\deg(v) = |\{u \in V \mid \{u, v\} \in E\}|$. +\end{definition} + +\begin{conjecture}[Erd\H{o}s-Gy\'{a}rf\'{a}s, 1995] +Si $G = (V, E)$ est un graphe avec $\delta(G) \ge 3$, alors $G$ contient un cycle $C_k$ de longueur $k$ tel que $k = 2^m$ pour un certain entier $m \ge 1$. +\end{conjecture} + +\section{Littérature Contextuelle et Analogies} +Le problème se situe à l'intersection de la théorie des graphes extrémaux et de la combinatoire additive. Il est structurellement similaire aux questions concernant l'existence de longueurs de cycles spécifiques dans les graphes denses (par exemple, les théorèmes de Bondy et Simonovits sur les cycles de longueur paire). + +Nous dressons une analogie avec la preuve de l'existence de cycles pairs dans des graphes de densité suffisante, où l'on construit typiquement un chemin maximal et étudie le voisinage de ses extrémités. Ici, nous cherchons à affiner le processus de construction du chemin en exploitant la condition sur le degré pour forcer des cycles de longueurs spécifiques. + +\section{Stratégie de Preuve et Lemmes} + +Nous décomposons la conjecture en sous-problèmes structurés. + +\begin{lemma}[Voisinage du Chemin Maximal] +\label{lem:longest_path} +Soit $P = v_1 v_2 \dots v_l$ un chemin maximal dans un graphe $G$ avec $\delta(G) \ge 3$. Alors tous les voisins de $v_1$ doivent se trouver sur $P$. +\end{lemma} + +\begin{proof} +Supposons qu'il existe un voisin $u$ de $v_1$ tel que $u \notin V(P)$. Alors le chemin $u v_1 v_2 \dots v_l$ a une longueur $l+1$, contredisant l'hypothèse selon laquelle $P$ est un chemin maximal. Ainsi, $N(v_1) \subseteq V(P)$. +\end{proof} + +\begin{lemma}[Formation de Cycles] +\label{lem:cycle_lengths} +Soit $P = v_1 v_2 \dots v_l$ un chemin maximal dans $G$ avec $\delta(G) \ge 3$. Soient $v_{i_1}, v_{i_2}, \dots, v_{i_k}$ les voisins de $v_1$ sur $P$, ordonnés tels que $2 = i_1 < i_2 < \dots < i_k \le l$. Puisque $\delta(G) \ge 3$, $k \ge 3$. Les arêtes $\{v_1, v_{i_j}\}$ créent des cycles $C^{(j)}$ de longueur $i_j$. +\end{lemma} + +\begin{proof} +Le segment de chemin $v_1 \dots v_{i_j}$ a une longueur $i_j - 1$. L'ajout de l'arête $\{v_1, v_{i_j}\}$ complète un cycle de longueur $(i_j - 1) + 1 = i_j$. +\end{proof} + +\section{Architecture de Formalisation Lean 4} +Pour autoformaliser cette preuve, nous esquissons les structures Lean 4. + +\begin{verbatim} +import Mathlib.Combinatorics.SimpleGraph.Basic +import Mathlib.Combinatorics.SimpleGraph.Connectivity + +variable {V : Type*} [Fintype V] [DecidableEq V] +variable (G : SimpleGraph V) + +def min_degree_at_least (k : \mathbb{N}) : Prop := + \forall v : V, G.degree v \ge k + +def has_cycle_power_of_two : Prop := + \exists (c : G.Walk v v), c.IsCycle \land \exists (m : \mathbb{N}), c.length = 2^m + +theorem erdos_gyarfas (h : min_degree_at_least G 3) : + has_cycle_power_of_two G := by + sorry +\end{verbatim} + +\section{Conclusion} +La décomposition structurelle fournie sert d'étape fondamentale vers la résolution de la conjecture d'Erd\H{o}s-Gy\'{a}rf\'{a}s, la préparant pour la vérification formelle. + +\end{document} diff --git 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