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Large triangle packings and Tuza’s conjecture in sparse random graphs

Part of: Graph theory Extremal combinatorics Designs and configurations

Published online by Cambridge University Press:  22 July 2020

Patrick Bennett ,
Andrzej Dudek  and
Shira Zerbib
Patrick Bennett
Affiliation:
Department of Mathematics, Western Michigan University, Kalamazoo, MI49008, USA
Andrzej Dudek*
Affiliation:
Department of Mathematics, Western Michigan University, Kalamazoo, MI49008, USA
Shira Zerbib
Affiliation:
Department of Mathematics, Iowa State University, Ames, IA50011, USA
*
*Corresponding author. Email: andrzej.dudek@wmich.edu
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Abstract

The triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.

MSC classification

Primary: 05B40: Packing and covering
Secondary: 05C80: Random graphs 05D40: Probabilistic methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 5 , September 2020 , pp. 757 - 779
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

Supported in part by Simons Foundation Grant #426894.

‡

Supported in part by Simons Foundation Grant #522400.

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