{
  "abstract": "Temporal graphs are graphs whose edge relation changes in discrete time steps. The Temporal Exploration Problem (TEXP) asks for a temporal walk that visits all vertices. We show that every connected temporal graph with $n$ vertices can be explored in $O(qn\\log^3(n))$ time steps, where $q$ is the number of vertex color classes induced by the 2-dimensional Weisfeiler-Leman (2-WL) coloring. For small $q$, our bound substantially improves on the known worst-case bound of $Θ(n^2)$ in general connected temporal graphs. A corresponding walk can be constructed in deterministic polynomial time. The upper bound is complemented by lower bounds of $Ω(n\\log(n))$ when $q$ is constant and $Ω(qn)$ when $1 \\leq q \\leq n$. We also apply our techniques to the Temporal Rendezvous Problem (TRP), in which two agents that do not communicate have to meet on a vertex of the temporal graph. Our exploration algorithm yields that, with only polynomial-time computational power, the agents can meet in $O(n\\log^3(n))$ time steps, leaving a gap of only $\\log^2(n)$ to the known lower bound. Dogeas, Erlebach, Kammer, Meintrup, and Moses Jr. [ICALP'24] have shown an $O(rn^{1+ε})$ bound for TEXP, for every fixed $ε> 0$, where $r$ is the number of orbits, and an $O(n^{1+ε})$ bound for TRP. Both results require superpolynomial computation time. Our contribution improves upon these results in several ways. First, the dependence on $n$ in the upper bounds on the number of time steps is decreased from $n^{1+ε}$ to $n\\log^3(n)$ for both TEXP and TRP. Second, for TEXP, we bound the number of time steps linearly by the parameter q, which is efficiently computable and upper bounded by $r$. Third, we provide novel deterministic polynomial-time algorithms for both TEXP and TRP.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30123",
  "author_total": 3,
  "authors": [
    {
      "name": "Frank Kammer"
    },
    {
      "name": "Sandra Kiefer"
    },
    {
      "name": "Johannes Meintrup"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d73c4y7h91896.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30123",
  "fulltext_url": null,
  "id": "d73c4y7h91896",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30123",
  "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.CC"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7223203617",
  "pdf_url": "https://arxiv.org/pdf/2610.30123",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d73c4y7h91896.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.30123",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30123"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Temporal Exploration and Rendezvous via Weisfeiler-Leman Colorings",
  "title_text": "Temporal Exploration and Rendezvous via Weisfeiler-Leman Colorings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/d73c4y7h91896",
  "venue": "arXiv",
  "versions": []
}