{
  "id": "2204.04479",
  "version": "v1",
  "published": "2022-04-09T14:21:02.000Z",
  "updated": "2022-04-09T14:21:02.000Z",
  "title": "On local antimagic vertex coloring for complete full $t$-ary trees",
  "authors": [
    "Martin Bača",
    "Andrea Semaničová--Feňovčíková",
    "Ruei-Ting Lai",
    "Tao-Ming Wang"
  ],
  "comment": "15 pages, 6 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G = (V, E)$ be a finite simple undirected graph without $K_2$ components. A bijection $f : E \\rightarrow \\{1, 2,\\cdots, |E|\\}$ is called a local antimagic labeling if for any two adjacent vertices $u$ and $v$, they have different vertex sums, i.e., $w(u) \\neq w(v)$, where the vertex sum $w(u) = \\sum_{e \\in E(u)} f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus any local antimagic labeling induces a proper vertex coloring of $G$ where the vertex $v$ is assigned the color (vertex sum) $w(v)$. The local antimagic chromatic number $\\chi_{la}(G)$ is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. It was conjectured \\cite{Aru-Wang} that for every tree $T$ the local antimagic chromatic number $l+ 1 \\leq \\chi_{la} ( T )\\leq l+2$, where $l$ is the number of leaves of $T$. In this article we verify the above conjecture for complete full $t$-ary trees, for $t \\geq 2$. A complete full $t$-ary tree is a rooted tree in which all nodes have exactly $t$ children except leaves and every leaf is of the same depth. In particular we obtain that the exact value for the local antimagic chromatic number of all complete full $t$-ary trees is $ l+1$ for odd $t$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-09T14:21:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ary tree",
      "local antimagic vertex coloring",
      "complete full",
      "local antimagic chromatic number",
      "local antimagic labeling"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}