{
  "id": "2310.20462",
  "version": "v1",
  "published": "2023-10-31T13:54:38.000Z",
  "updated": "2023-10-31T13:54:38.000Z",
  "title": "Anti-van der Waerden Numbers of Graph Products with Trees",
  "authors": [
    "Zhanar Berikkyzy",
    "Joe Miller",
    "Elizabeth Sprangel",
    "Shanise Walker",
    "Nathan Warnberg"
  ],
  "comment": "20 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a graph $G$, an exact $r$-coloring of $G$ is a surjective function $c:V(G) \\to [1,\\dots,r]$. An arithmetic progression in $G$ of length $j$ with common difference $d$ is a set of vertices $\\{v_1,\\dots, v_j\\}$ such that $dist(v_i,v_{i+1}) = d$ for $1\\le i < j$. An arithmetic progression is rainbow if all of the vertices are colored distinctly. The fewest number of colors that guarantees a rainbow arithmetic progression of length three is called the anti-van der Waerden number of $G$ and is denoted $aw(G,3)$. It is known that $3 \\le aw(G\\square H,3) \\le 4$. Here we determine exact values $aw(T\\square T',3)$ for some trees $T$ and $T'$, determine $aw(G\\square T,3)$ for some trees $T$, and determine $aw(G\\square H,3)$ for some graphs $G$ and $H$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-31T13:54:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C15",
      "05C12",
      "G.2.2",
      "G.2.1"
    ],
    "keywords": [
      "anti-van der waerden number",
      "graph products",
      "determine exact values",
      "rainbow arithmetic progression"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}