关于图的线性荫度一个猜想的研究

The linear arboricity of a graph is the minimum number of linear forests which decompose the edge set of the graph. The theory of linear arboricity has been a research concern in the field of coloring of graphs since its establishment. Currently, there exist a lot of research results.This project will focus on the linear arboricity conjecture (LAC) and the linear arboricity conjecture of planar graphs. By studying the structure of graphs deeply, we will obtain some useful properties to solve the above problem. At the same time, we will focus on cosidering the three following aspects: 1.Determining the linear arboricity of planar graphs with maximum degree at leaat seven. 2. Determining the linear arboricity of graphs on surfaces with maximum degree at least nine. 3.Studying the linear arboricity of planar graphs with maximum degree at least five. In recent years, the linear arboricity conjecture，in particular the linear arboricity conjecture of planar graphs is one of the hot points which concerned by graph theory scholars. Any progress in these issues can attract domestic and foreign scholars'' attention, so our topic is deserve to research, and has good development prospects.

图的线性荫度是指把一个图的边集合分解成线性森林的最少个数。自线性荫度理论建立以来就成为图的染色领域备受关注的研究方向，目前已经有大量的研究成果。本项目将围绕解决线性荫度猜想（LAC）和平面图的线性荫度猜想展开研究。通过深入研究图的结构，得到一些可以利用的性质来解决上述相关的问题。与此同时，本项目将着重考虑如下三个方面：1、确定最大度至少为7的平面图的线性荫度；2、确定最大度至少为9的曲面图的线性荫度；3、研究最大度至少为5的平面图的线性荫度。最近几年，图的线性荫度猜想，特别是平面图的线性荫度猜想是受到图论学者关注的热点之一，关于这些问题的任何进展都能吸引国内外学者关注，所以我们的课题是有研究价值的，并具有良好的发展前景。

本项目给出以下结果：1.确定了最大度至少是7，不含有相邻短圈的平面图的线性荫度；2.确定了最大度至少是7，不含有含弦的5-圈的平面图的线性荫度；3.给出了最大度至少是9，可嵌入到欧拉示性数非负的曲面图的线性荫度；4.给出了不含有含弦的5-，6-圈的平面图和不含有相交的4-圈的平面图的线性2-荫度的一个上界。共发表论文两篇（SCI），投稿论文3篇，圆满完成了本项目的研究。