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Tuza's Conjecture is Asymptotically Tight for Dense Graphs

Part of: Graph theory

Published online by Cambridge University Press:  14 March 2016

JACOB D. BARON  and
JEFF KAHN
JACOB D. BARON
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ, USA (e-mail: jabar@math.rutgers.edu, jkahn@math.rutgers.edu)
JEFF KAHN
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ, USA (e-mail: jabar@math.rutgers.edu, jkahn@math.rutgers.edu)
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Abstract

An old conjecture of Z. Tuza says that for any graph G, the ratio of the minimum size, τ3(G), of a set of edges meeting all triangles to the maximum size, ν3(G), of an edge-disjoint triangle packing is at most 2. Here, disproving a conjecture of R. Yuster, we show that for any fixed, positive α there are arbitrarily large graphs G of positive density satisfying τ3(G) > (1 − o(1))|G|/2 and ν3(G) < (1 + α)|G|/4.

MSC classification

Primary: 05C70: Factorization, matching, partitioning, covering and packing
Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 5 , September 2016 , pp. 645 - 667
Copyright
Copyright © Cambridge University Press 2016 

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