{
  "id": "2209.14176",
  "version": "v1",
  "published": "2022-09-28T15:42:11.000Z",
  "updated": "2022-09-28T15:42:11.000Z",
  "title": "Homogeneous Sets in Graphs and a Chromatic Multisymmetric Function",
  "authors": [
    "Logan Crew",
    "Evan Haithcock",
    "Josephine Reynes",
    "Sophie Spirkl"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-28T15:42:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05C15"
    ],
    "keywords": [
      "chromatic multisymmetric function",
      "homogeneous sets",
      "chromatic symmetric function",
      "unit interval graphs",
      "vertex set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}