色对称函数理论于1995年由美国科学院院士Richard Stanley最先开始研究，现已成为组合数学研究的一个重要课题，与群表示论、代数几何等多个数学分支都有密切的联系。本项目将围绕Stanley的(3+1)-free猜想和Stanley的树同构猜想研究(3+1)-free的偏序集的不可比图、无爪图以及树的色对称函数的组合性质，主要包括：(3+1)-free的偏序集的不可比图的色对称函数的初等对称函数展开式中系数的非负性，无爪图的色对称函数的Schur函数展开式中系数的非负性，以及这两类图的独立多项式的实根性的新证明，以及利用色对称函数研究对顶点的度加以限制的树的同构问题。项目预期将在Stanley的(3+1)-free猜想和Stanley的树同构猜想方面取得一些进展，为最终解决这两个猜想提供新的思路，进一步丰富色对称函数理论。

The theory of chromatic symmetric functions was initially studied by Richard Stanley in 1995, and now it is a very important subject of symmetric functions, which has very close relationship with other mathematical branches such as representation theory and algebraic geometry. Motivated by Stanley’s (3+1)-free conjecture and Stanley's isomorphism conjecture for trees, in this project we will focus on the study of the combinatorics of the chromatic symmetric functions of the following three kinds of graphs: incomparability graphs of (3+1)-free posets, claw-free graphs, and trees. Precisely, we will study non-negativity of the coefficients in the elementary symmetric function expansion of the chromatic symmetric functions of incomparability graphs of (3+1)-free posets, non-negativity of the coefficients in the Schur function expansion of the chromatic symmetric functions of claw-free posets, new proofs of the real-rootedness of their independence polynomials, and the isomorphism of trees with restricted vertex-degree condition by using their chromatic symmetric functions. We hope that some progress can be achieved on Stanley’s (3+1)-free conjecture and Stanley's isomorphism conjecture for trees. We think that our progress will provide new clues for finally solving these two conjectures, and will enhance the development of the theory of chromatic symmetric functions.