路与圈Mycielski图的线性荫度

The Linear Arboricity of Mycielski Graph of Path and Cycle

摘要: 将图$G$ 的边集$E\left(G\right)$ 通过划分，分解为$m$ 个边互不交的线性森林时，所需要线性森林最小的数目$m$ 称为图的线性荫度，此概念由Harary于1970年首次提出，它丰富了图的理论，又拓展了图的应用领域。本文针对由路与圈构造而成的Mycielski图开展探讨，按照构造图的结构特点，通过对其边进行分解，证明了路与圈的Mycielski图满足线性荫度的猜想。

Abstract: When the edge set $E\left(G\right)$ of graph $G$ is partitioned into $m$ mutually disjoint linear forests, the minimum number $m$ required is termed the linear arboricity of the graph. This concept was first introduced by Harary in 1970, enriching graph theory and expanding its applications. This paper investigates Mycielski graphs constructed from paths and cycles. By analyzing their structural characteristics and decomposing their edges, it is proved that Mycielski graphs composed of paths and cycles satisfy the linear arboricity conjecture.