Paths of Length Three are $K_{r+1}$-Turán-Good

The generalized Turán problem ex$(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Turán problem is often the original Turán graph. They gave the name "$F$-Turán-good" to graphs $T$ for which, for large enough $n$, the solution to the generalized Turán problem is realized by a Turán graph. They prove that the path graph on two edges, $P_2$, is $K_{r+1}$-Turán-good for all $r \ge 3$, but they conjecture that the same result should hold for all $P_\ell$. In this paper, using arguments based in flag algebras, we prove that the path on three edges, $P_3$, is also $K_{r+1}$-Turán-good for all $r \ge 3$.

Sharp bounds for decomposing graphs into edges and triangles

For a real constant α, let $\pi _3^\alpha (G)$ be the minimum of twice the number of K2’s plus α times the number of K3’s over all edge decompositions of G into copies of K2 and K3, where Kr denotes the complete graph on r vertices. Let $\pi _3^\alpha (n)$ be the maximum of $\pi _3^\alpha (G)$ over all graphs G with n vertices. The extremal function $\pi _3^3(n)$ was first studied by Győri and Tuza (Studia Sci. Math. Hungar.22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova (Combin. Probab. Comput.28 (2019) 465–472) proved via flag algebras that $\pi _3^3(n) \le (1/2 + o(1)){n^2}$ . We extend their result by determining the exact value of $\pi _3^\alpha (n)$ and the set of extremal graphs for all α and sufficiently large n. In particular, we show for α = 3 that Kn and the complete bipartite graph ${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$ are the only possible extremal examples for large n.

Minimizing the number of 5-cycles in graphs with given edge-density

Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle C5. We show that every graph of order n and size $ (1 - 1/k) \left( {\matrix{n \cr 2 }} \right) $, where k ≥ 3 is an integer, contains at least $$({1 \over {10}} - {1 \over {2k}} + {1 \over {{k^2}}} - {1 \over {{k^3}}} + {2 \over {5{k^4}}}){n^5} + o({n^5})$$ copies of C5. This bound is optimal, since a matching upper bound is given by the balanced complete k-partite graph. The proof is based on the flag algebras framework. We also provide a stability result. An SDP solver is not necessary to verify our proofs.

On the Caccetta-Häggkvist Conjecture with a Forbidden Transitive Tournament

The Caccetta-Häggkvist Conjecture asserts that every oriented graph on $n$ vertices without directed cycles of length less than or equal to $l$ has minimum outdegree at most $(n-1)/l$. In this paper we state a conjecture for graphs missing a transitive tournament on $2^k+1$ vertices, with a weaker assumption on minimum outdegree. We prove that the Caccetta-Häggkvist Conjecture follows from the presented conjecture and show matching constructions for all $k$ and $l$. The main advantage of considering this generalized conjecture is that it reduces the set of the extremal graphs and allows using an induction.We also prove the triangle case of the conjecture for $k=1$ and $2$ by using the Razborov's flag algebras. In particular, it proves the most interesting and studied case of the Caccetta-Häggkvist Conjecture in the class of graphs without the transitive tournament on 5 vertices. It is also shown that the extremal graph for the case $k=2$ has to be a blow-up of a directed cycle on 4 vertices having in each blob an extremal graph for the case $k=1$ (complete regular bipartite graph), which confirms the conjectured structure of the extremal examples.