标题： 基于图分类的陈-拉帕萨德猜想的模块化归纳证明 显示英文标题

标题： A Modular Inductive Proof of the Chen-Raspaud Conjecture via Graph Classification

摘要： 猜想由陈和拉施帕德提出，对于每个整数$k \ge 2$，任何具有\[ \mathrm{mad}(G) < \frac{2k+1}{k} \quad\text{and}\quad \mathrm{odd\text{-}girth}(G) \ge 2k+1 \]的图$G$都可以同态映射到Kneser图$K(2k+1,k)$。 基本情况$k=2$和$k=3$从以前的工作中已知。 这里提供了一个模归纳证明，其中在层次$k+1$的图被分为四种类结构，并通过禁止配置消除、充电论证、路径坍缩技术以及将较小的Kneser图组合嵌入到较大的Kneser图中的组合嵌入方法，证明它们没有最小反例。 这个论证完成了对所有$k \ge 2$的归纳，从而在一般性上解决了Chen-Raspaud猜想。 显示英文摘要

摘要： It is conjectured by Chen and Raspaud that for each integer $k \ge 2$, any graph $G$ with \[ \mathrm{mad}(G) < \frac{2k+1}{k} \quad\text{and}\quad \mathrm{odd\text{-}girth}(G) \ge 2k+1 \] admits a homomorphism into the Kneser graph $K(2k+1,k)$. The base cases $k=2$ and $k=3$ are known from earlier work. A modular inductive proof is provided here, in which graphs at level $k+1$ are classified into four structural classes and are shown to admit no minimal counterexamples by means of forbidden configuration elimination, a discharging argument, path-collapsing techniques, and a combinatorial embedding of smaller Kneser graphs into larger ones. This argument completes the induction for all $k \ge 2$, thus settling the Chen-Raspaud conjecture in full generality.