{
  "abstract": "A majority coloring of a graph $G$ is a vertex coloring of $G$ in which no vertex has more than half of its neighbors colored with its own color. The least number of colors required for a majority coloring of $G$ is the majority chromatic number $μ(G)$. The majority coloring game, introduced by Bosek et al. (2019), is a two-player Maker-Breaker-type game where the players alternately color vertices while maintaining the majority condition at each vertex; the least number of colors required for the first player to have a winning strategy on $G$ is the majority game chromatic number $μ_g(G)$. In contrast with the static case, Bosek et al. show that $μ_g(G)$ is unbounded in general, while $μ_g(G)\\le col_g(G)$, where $col_g(G)$ is the game coloring number of $G$. It is known (cf. Faigle et al. (1993)) that, for any acyclic graph $G$, $col_g(G) \\leq 4$, so $μ_g(G) \\leq 4$ as well. We improve this bound by showing that $μ_g(G) \\leq 3$ for any acyclic graph $G$ of maximum degree at most $4$. We also show that $μ_g(G) = 2$ if $G$ is a nonempty path, star, or complete graph, or a disjoint union of such graphs. These improve the results of Bosek et al. We also initiate the study of the computational complexity of the pre-coloring extension problem for the static and game versions of majority coloring. We show that the Majority $k$-precoloring Extension problem is NP-complete for each $k \\geq 2$. When $k = 2$, it remains NP-complete on the class of bipartite graphs where maximum degrees of each part are $5$ and $6$, and when $k = 3$ it remains NP-complete on the class of $12$-regular planar graphs. We show that the game version is PSPACE-complete, even when the number of colors $k=χ(G)$, and even when $k=2$ and $G$ is a bipartite graph in which one part has maximum degree equal to $9$. The results on the game versions also hold for the variation where Bob starts the game.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23803",
  "author_total": 3,
  "authors": [
    {
      "name": "Yash Chawda"
    },
    {
      "name": "Saraswati Girish Nanoti"
    },
    {
      "name": "Brahadeesh Sankarnarayanan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/38trm5q6gh6fb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23803",
  "fulltext_url": null,
  "id": "38trm5q6gh6fb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23803",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214047497",
  "pdf_url": "https://arxiv.org/pdf/2609.23803",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/38trm5q6gh6fb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2609.23803",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23803"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the majority game chromatic number of forests and other graphs",
  "title_text": "On the majority game chromatic number of forests and other graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/38trm5q6gh6fb",
  "venue": "arXiv",
  "versions": []
}