{
  "abstract": "An edge labeling of a connected graph $G=(V, E)$ is called a local antimagic labeling if it is a bijection $f: E\\rightarrow \\{1, 2, \\cdots, \\left | E \\right | \\}$ such that for any pair of adjacent vertices $u$ and $v$, $f^{+}(u)\\neq f^{+}(v)$ where $f^{+}(x)=\\sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted $χ_{la}(G)$, is the minimum number of distinct induced vertex labels across all possible local antimagic labelings of $G$. This paper investigates two families of graphs $G(m^{[n]})$ (a graph class constructed by augmenting a bipartite structure $B(m^{[n]})$ with cyclic edges) and $K_{1,s}\\times C_n$ (the direct product of a star $K_{1,s}$ and a cycle $C_n$). Under the conditions odd integer $m\\ge 1$ and even integer $n\\ge 4$, we establish the exact values of $χ_{la}(G(m^{[n]}))$ and $χ_{la}(K_{1,s}\\times C_n)$ for various $s$. Furthermore, we characterize several graph families achieving $χ_{la}(G)=2$ or $3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14486",
  "author_total": 3,
  "authors": [
    {
      "name": "Huimin Xu"
    },
    {
      "name": "Gee-Choon Lau"
    },
    {
      "name": "Ruixue Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8hhaax030gqw8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14486",
  "fulltext_url": null,
  "id": "8hhaax030gqw8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14486",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7221762149",
  "pdf_url": "https://arxiv.org/pdf/2610.14486",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8hhaax030gqw8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14486",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14486"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Local Antimagic Chromatic Number of Middle Graph and Direct Product of Graphs",
  "title_text": "Local Antimagic Chromatic Number of Middle Graph and Direct Product of Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/8hhaax030gqw8",
  "venue": "arXiv",
  "versions": []
}