论文标题
图和色度对称函数中的均匀集
Homogeneous Sets in Graphs and a Chromatic Multisymmetric Function
论文作者
论文摘要
在本文中，我们将色度对称函数$ x $扩展到一个色度$ k $ -multisymmetric $ x_k $，该函数定义为配备了其顶点设置的划分为$ k $零件的图形。我们证明，此新功能保留了$ x $的基本属性和基础扩展，并且我们通过通过$ x_k $传递$ x $的$ x $从系统地衍生出新的线性关系的方法。 特别是，我们展示了如何利用$ g $的均匀套件（这些$ s \ subseteq v（g）$，以便每个顶点的$ v（g）\ backslash s $要么与$ s $的所有顶点相邻，要么与$ s $的所有顶点相邻），以与$ g $ g $ thy simple of simple of simple of simple of simple of simple of simplesmertric相关联。此外，我们展示了如何将这个想法扩展到均质对$ s_1 \ sqcup S_2 \ subseteq v（g）$概括了Guay-Paquet使用的过程，以将Stanley-STAMBRIDGE Suntienture降低到单位间隔图。
In this paper, we extend the chromatic symmetric function $X$ to a chromatic $k$-multisymmetric function $X_k$, defined for graphs equipped with a partition of their vertex set into $k$ parts. We demonstrate that this new function retains the basic properties and basis expansions of $X$, and we give a method for systematically deriving new linear relationships for $X$ from previous ones by passing them through $X_k$. In particular, we show how to take advantage of homogeneous sets of $G$ (those $S \subseteq V(G)$ such that each vertex of $V(G) \backslash S$ is either adjacent to all of $S$ or is nonadjacent to all of $S$) to relate the chromatic symmetric function of $G$ to those of simpler graphs. Furthermore, we show how extending this idea to homogeneous pairs $S_1 \sqcup S_2 \subseteq V(G)$ generalizes the process used by Guay-Paquet to reduce the Stanley-Stembridge conjecture to unit interval graphs.
