arXiv subjects
Publications and source records attributed to Michał Fiedorowicz.
This document collects several approaches attempted in proving the Chen-Raspaud Conjecture for k = 3. Each approach is detailed with full proofs and explanations of why it failed. This compilation aims to provide insights and a foundation for future research on this conjecture.
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.
In this paper, the Painlevé property to fractional differential equations (FDEs) are extended and the existence and uniqueness theorems for both linear and nonlinear FDEs are established. The results contribute to the research of integrability and solvability in the context of fractional calculus, which has significant implications in various fields such as physics, engineering, and applied sciences. By bridging the gap between pure mathematical theory and practical applications, this work provides a foundational understanding that can be utilized in modeling phenomena exhibiting memory and hereditary properties.