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Immersions and Albertson’s Conjecture

Authors Jacob Fox, János Pach, Andrew Suk

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • Immersions
  • crossing numbers
  • chromatic number

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Abstract

A graph is said to contain K_k (a clique of size k) as a weak immersion if it has k vertices, pairwise connected by edge-disjoint paths. In 1989, Lescure and Meyniel made the following conjecture related to Hadwiger’s conjecture: Every graph of chromatic number k contains K_k as a weak immersion. We prove this conjecture for graphs with at most 1.4(k-1) vertices. As an application, we make some progress on Albertson’s conjecture on crossing numbers of graphs, according to which every graph G with chromatic number k satisfies cr(G) ≥ cr(K_k). In particular, we show that the conjecture is true for all graphs of chromatic number k, provided that they have at most 1.4(k-1) vertices and k is sufficiently large.

Cite As Get BibTex

Jacob Fox, János Pach, and Andrew Suk. Immersions and Albertson’s Conjecture. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 50:1-50:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.50

Author Details

Jacob Fox
  • Department of Mathematics, Stanford University, CA, USA
János Pach
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Andrew Suk
  • Department of Mathematics, University of California San Diego, La Jolla, CA, USA

Funding

  • Fox, Jacob: supported by NSF Award DMS-2154129.
  • Pach, János: Rényi Institute, Budapest. Supported by NKFIH grant K-176529 and ERC Advanced Grant "GeoScape."
  • Suk, Andrew: Supported by NSF awards DMS-1952786 and DMS-2246847.

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