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

TL;DR

This paper proves the Chen-Raspaud conjecture for all integers k ≥ 2 by classifying graphs into structural classes and using inductive, combinatorial, and discharging techniques to eliminate counterexamples.

Contribution

It provides a modular inductive proof that confirms the Chen-Raspaud conjecture for all relevant graph classes, completing the proof for all k ≥ 2.

Findings

The conjecture holds for all k ≥ 2.

Graphs with specified mad and odd-girth admit homomorphisms into Kneser graphs.

The proof introduces a classification and elimination method for potential counterexamples.

Abstract

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.