{
  "abstract": "The graph coloring game is a two-player game in which the players alternately color an uncolored vertex of a graph $G$. The game chromatic number is the minimum number of colors needed for the first player to guarantee a win. We investigate this parameter for generalized Mycielski graphs $M_k(G)$, where $G$ is a path $P_n$ or a cycle $C_n$ with $n$ vertices. For every $k\\geq2$ and $n\\geq5$, we establish $4\\leqχ_g\\bigl(M_k(P_n)\\bigr)\\leq5$ and $4\\leqχ_g\\bigl(M_k(C_n)\\bigr)\\leq5$. We also determine the exact values $χ_g\\bigl(M_2(P_5)\\bigr)=χ_g\\bigl(M_2(P_6)\\bigr)=4$. The proofs of the lower bounds use a configuration in which Bob can create two threats simultaneously, while the four-color upper bounds in the two exact cases are proved using the double-doctor lemma. Thus the number of layers and the order of the base graph may grow, but the game chromatic number remains bounded by five.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02283",
  "author_total": 3,
  "authors": [
    {
      "name": "Yushuang Mou"
    },
    {
      "name": "Qiang Sun"
    },
    {
      "name": "Chao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/687a8zr8w711v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02283",
  "fulltext_url": null,
  "id": "687a8zr8w711v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02283",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7207759168",
  "pdf_url": "https://arxiv.org/pdf/2609.02283",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/687a8zr8w711v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02283",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02283"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The game chromatic number of generalized Mycielski graphs of paths and cycles",
  "title_text": "The game chromatic number of generalized Mycielski graphs of paths and cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/687a8zr8w711v",
  "venue": "arXiv",
  "versions": []
}