{
  "abstract": "We study weighted edge coloring, where each edge of a graph $G$ has a weight in $(0,1]$, and colors are assigned so that the incident weight of each color at every vertex is at most one. The objective is to minimize the number of colors, denoted by $χ'_w(G)$. This generalizes classical edge coloring (all weights are one) and bin packing ($G$ is a star). Two natural lower bounds are the maximum weighted degree $Δ_w(G)$ and the maximum local bin-packing number $\\mathrm{bp}(G)$: the maximum, over vertices, of the minimum number of unit bins needed to pack the incident weights. We have $\\lceilΔ_w(G)\\rceil\\le\\mathrm{bp}(G)\\leχ'_w(G)$. How closely these local lower bounds approximate the global optimum is a longstanding open problem. Our main result establishes an asymptotic weighted analogue of Vizing's theorem: every simple graph admits a coloring using at most $(1+o(1))\\mathrm{bp}(G)$ colors. Thus local bin packings can be made globally consistent with asymptotically vanishing relative loss, even in nonbipartite graphs. Our local-to-global rounding framework coordinates feasible local packing distributions into a partial coloring, leaving only $o(\\mathrm{bp}(G))$ uncolored weight at every vertex. For bipartite multigraphs, we prove $χ'_w(G)\\le(2+o(1))Δ_w(G)$, establishing the Chung--Ross conjecture [SICOMP'91] asymptotically. For loopless multigraphs, we obtain the exact bound $χ'_w(G)\\le3\\lceilΔ_w(G)\\rceil$. Both leading coefficients are optimal, giving weighted analogues of the classical Kőnig--Shannon bounds. Further upper and lower bounds show that the classical factor-$3/2$ separation does not persist with the local bin-packing benchmark. We also disprove the exact weighted-degree conjecture of Feige and Singh [ESA'08]. Our upper bounds have polynomial-time algorithms, and their existence statements are formalized in Lean 4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03190",
  "author_total": 2,
  "authors": [
    {
      "name": "Rishikesh Gajjala"
    },
    {
      "name": "Arindam Khan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/500s0nwbfb306.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03190",
  "fulltext_url": null,
  "id": "500s0nwbfb306",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03190",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03190",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/500s0nwbfb306.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03190",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03190"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Asymptotically Optimal Weighted Edge Colorings via Local Bin Packing",
  "title_text": "Asymptotically Optimal Weighted Edge Colorings via Local Bin Packing",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/500s0nwbfb306",
  "venue": "arXiv",
  "versions": []
}