{
  "abstract": "A majority coloring of a graph $G = (V,E)$ is a coloring of $V(G)$ such that, for each vertex $v$, the number of neighbors of $v$ with the same color as $v$ is at most $deg(v)/2$. A strong majority coloring is a coloring of $V(G)$ such that, for each vertex $v$, every monochromatic subset of $N(v)$ has size at most $deg(v)/2$. The (strong) majority coloring game is a two-player Maker-Breaker-type game, in which two players Alice and Bob color the vertices of a graph $G$ alternately, maintaining the (strong) majority condition. The least number of colors such that Alice has a winning strategy in such a game is called the (strong) majority game chromatic number of the graph $G$, denoted $μ_g(G)$ (or $\\mathrm{Maj}_g(G)$ for the strong version). For the majority coloring game, we prove that $μ_g(G) \\le 3$ under the following cases: $G$ is a $2$-caterpillar, $G$ is a rooted tree with all leaves at depth $k \\le 4$, and $G$ is a subdivision of some graph. The latter resolves a problem posed by Bosek--Grytczuk--Jakóbczak in 2019, who also asked whether $μ_g(T) \\le 3$ for every tree $T$. For the latter question, we discuss various difficulties that arise when natural strategies are attempted by Alice to win the majority coloring game on trees. We include a comparison with the marking game and relaxed coloring game on trees, and with the majority coloring game on locally finite acyclic graphs $G$ with $δ(G) > 1$. For the strong majority coloring game, we compute $\\mathrm{Maj}_g(C_n)$ exactly for each cycle $C_n$, $n \\ge 3$. We also initiate the study of the computational complexity of the strong majority coloring game; specifically, we prove that the decision version of the Strong Majority Game Chromatic Number problem is PSPACE-complete. We also show that the Strong Majority 2-Coloring problem is NP-complete on Eulerian graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37796",
  "author_total": 4,
  "authors": [
    {
      "name": "Yash Chawda"
    },
    {
      "name": "Saraswati Girish Nanoti"
    },
    {
      "name": "Brahadeesh Sankarnarayanan"
    },
    {
      "name": "Eshwar Srinivasan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9gcx2xmwbnx6r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37796",
  "fulltext_url": null,
  "id": "9gcx2xmwbnx6r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37796",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.37796",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9gcx2xmwbnx6r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37796",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37796"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Structural and computational aspects of majority coloring games",
  "title_text": "Structural and computational aspects of majority coloring games",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/9gcx2xmwbnx6r",
  "venue": "arXiv",
  "versions": []
}