On Tuza’s Conjecture in Dense Graphs
[August, 2016] Rutgers graduate student Jake
Baron and his advisor Jeff Kahn

Jake Baron and Jeff Kahn at Baron's dissertation defense 
have
provided a construction [1] that shows that a bound on the size of
minimum triangle edge cover of a graph G conjectured by Zsolt Tuza
[2] in 1981 is, in fact, asymptotically tight for an infinite family
of dense graphs. This disproves a more recent conjecture to the
contrary by Raphael Yuster [3].
A triangle edge cover in the graph G is a set of edges of G such
that every triangle of G contains an edge in the set. A minimum
triangle edge cover in G is a triangle edge cover containing the
fewest possible edges. Tuza’s conjecture relates the size of a
minimum triagle edge cover in G to the size of a largest triangle
packing, where a triangle packing is a set of triangles in G that
are pairwise edge disjoint.
If we let C(G) denote the size of a minimum triangle edge cover and
let P(G) denote the size of a maximum triangle packing in G, then it
is relatively straightforward to see that P(G) ≤ C(G) ≤ 3P(G). The
lower bound follows from the observation that at least one edge must
be deleted from every triangle in the packing to obtain a
trianglefree graph, while the upper bound follows because deleting
every edge from every triangle in the packing would certainly
eliminate all triangles from the graph.
Tuza’s conjecture is that C(G) ≤ 2P(G).
Tuza’s bound is tight for the complete
graphs on four and five nodes. For the complete graph on four nodes,
C = 2 and P = 1, while on five nodes C = 4 and P = 2. The latter is
illustrated in the figure below.
Tuza’s bound remains tight for certain arbitrarily large graphs
constructed by chaining together copies of the complete graph on
four nodes in prescribed ways, but its status is open for more
general large graphs. The best bound for general graphs (due to
Penny Haxell [4]) assures that C(G) ≤ (66/23)P(G)—closer to the
trivial bound than that conjectured by Tuza. On the other hand,
results noted by Yuster [3] assure that Tuza’s bound holds
asymptotically in graphs with quadratically large triangle covers.
Specifically, if G is a graph containing n nodes for which C(G) ≥ b
n^{2} for any fixed b > 0, then C(G) < (2+o(1)) P(G). Yuster then
conjectured that the factor of 2 in Tuza’s conjecture is not, in
fact, optimal in dense graphs in which the minimum triangle edge
covers approach using half the edges in G (which is essentially as
large as they could possibly be).
In light of Yuster’s conjecture, the question becomes whether the
“2” in Tuza’s bound is tight for an infinite family of graphs
containing quadraticallymany triangles, or whether it could be
improved (to say “3/2”) as the size of such a graph tends to
infinity.
The construction that Baron and Kahn devised settles this question
and disproves Yuster’s conjecture. It provides an infinite family of
arbitrarily large dense graphs in which C(G) asymptotically
approaches half the number of edges in G, and for which Tuza’s
conjecture is asymptotically tight. The proof is rather involved,
but their partly random construction ultimately guarantees, for any
positive a, arbitrarily large graphs G of sufficient density that
satisfy C(G) > (1o(1)) G/2 and P(G) < (1+a)G/4.
These results are part of Baron’s PhD dissertation entitled “Two
Problems on Cycles in Random Graphs” which he successfully defended
on August 31, 2016. The results were also presented by Kahn at the DIMACS
Workshop in Honor of Alan Hoffman. Baron received a 3year
fellowship associated with CCICADA,
the homeland security center based at DIMACS, to support his
graduate studies in mathematics. The fellowship was funded by a
grant to DIMACS/CCICADA from the Department of Homeland Security.
A version of [1] is available on the arXiv: [arXiv paper]
Printable version of this story: [PDF]
References:
[1] J. D. Baron and J. Kahn, “Tuza’s Conjecture is Asymptotically
Tight for Dense Graphs,” Combinatorics, Probability & Computing
25(5): 645667 (2016).
[2] Z. Tuza, “Conjecture,” in A. Hajnal, L. Lovász, and V. T. Sós
(eds.), Finite and Infinite Sets, Proc. Colloq. Math. Soc. Janos
Bolyai, page 888. NorthHolland, Amsterdam, 1981.
[3] R. Yuster, “Dense Graphs with a Large Triangle Cover Have a
Large Triangle Packing,” Combinatorics, Probability & Computing,
21(6):952962, (2012).
[4] P. Haxell, “Packing and Covering Triangles in Graphs,” Discrete
Mathematics, 195:251254, Jan 1999.
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