{
  "abstract": "A famous folklore result in list coloring demonstrating that the gap between the list chromatic number and chromatic number of a graph can be arbitrarily large is: $χ_{\\ell}(K_{l,t}) = 1+l$ if and only if $t \\geq l^l$. DP-coloring (also called correspondence coloring) is a well-studied generalization of list coloring introduced in 2015. In 2018, Mudrock studied the DP analogue of the aforementioned folklore result. He proved that for $l \\in\\mathbb{N}$, if $μ(l)$ is the smallest integer $t$ such that $χ_{DP}(K_{l,t})=1+l$, then $\\left\\lceil l^l/l!\\right\\rceil \\leq μ(l) \\leq 1+l^l(\\log(l!)+1)/l!$. Recently, Kaul, Mudrock, and Sharma studied a more general version of this problem by studying the smallest $t$ for which $χ_{DP}(G \\square K_{l,t}) = k + l$, where $G$ satisfies certain criticality conditions and $G \\square K_{l,t}$ denotes the Cartesian product of $G$ and $K_{l,t}$. In this paper, we introduce a notion we call covering families that gives a new perspective on these DP-coloring questions. In particular, if $κ(l)$ denotes the minimum size of a covering family of $[l]^l$, we show that $μ(l)=κ(l)$. We use this equivalence to prove $μ(4)=12$ and to obtain new general lower bounds on $μ(l)$. We also prove a general upper bound on the minimum size of covering families which yields an improved general upper bound on $μ(l)$ and gives improvements on known bounds for related DP-coloring questions involving Cartesian products with complete bipartite graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33966",
  "author_total": 6,
  "authors": [
    {
      "name": "Hemanshu Kaul"
    },
    {
      "name": "Jeffrey A. Mudrock"
    },
    {
      "name": "Emily A. Psyhogios"
    },
    {
      "name": "Gunjan Sharma"
    },
    {
      "name": "Illia Siutkin"
    },
    {
      "name": "Aparna Upadhyay"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6ewtstzn3e2qk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33966",
  "fulltext_url": null,
  "id": "6ewtstzn3e2qk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33966",
  "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": "W7214702368",
  "pdf_url": "https://arxiv.org/pdf/2609.33966",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6ewtstzn3e2qk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33966",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33966"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Covering Families for DP-Coloring of Cartesian Products with Complete Bipartite Graphs",
  "title_text": "Covering Families for DP-Coloring of Cartesian Products with Complete Bipartite Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/6ewtstzn3e2qk",
  "venue": "arXiv",
  "versions": []
}