Jan Hladký

On the number of pentagons in triangle-free graphs

Using the formalism of flag algebras, we prove that every triangle-free graph G with n vertices contains at most (n/5)^5 cycles of length five. Moreover, the equality is attained only when n is divisible by five and G is the balanced blow-up of the pentagon. We also compute the maximal number of pentagons and characterize extremal graphs in the non-divisible case provided n is sufficiently large. This settles a conjecture made by Erdos in 1984.

Rank of divisors on tropical curves

We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, we confirm a conjecture of Baker asserting that the rank of a divisor D on a (non-metric) graph is equal to the rank of D on the corresponding metric graph, and construct an algorithm for computing the rank of a divisor on a tropical curve.

Brooks' Theorem via the Alon-Tarsi Theorem

We give a proof of Brooks’ theorem and its list coloring extension using the algebraic method of Alon and Tarsi; this also shows that the Brooks’ theorem remains valid in a more general game coloring setting.

AN APPROXIMATE VERSION OF THE TREE PACKING CONJECTURE

We prove that for any pair of constants ɛ > 0 and Δ and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most Δ, and with at most (n2) edges in total packs into

Loebl-Komlós-Sós Conjecture: dense case

{K_{(1 + \varepsilon )n}}

The Approximate Loebl--Komlós--Sós Conjecture III: The Finer Structure of LKS Graphs

. This implies asymptotic versions of the Tree Packing Conjecture of Gyárfás from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.

The Approximate Loebl--Komlós--Sós Conjecture II: The Rough Structure of LKS Graphs

Motivated by the Caccetta-Häggkvist Conjecture, we prove that every digraph on n vertices with minimum outdegree 0:3465n contains an oriented triangle. This improves the bound of 0:3532n of Hamburger, Haxell and Kostochka. The main new tool we use in our proof is the theory of flag algebras developed recently by Razborov.

The Approximate Loebl--Komlós--Sós Conjecture IV: Embedding Techniques and the Proof of the Main Result

Abstract We prove a version of the Loebl-Komlos-Sos Conjecture for large dense graphs. For any q > 0 there exists n 0 ∈ N such that for any n > n 0 holds: If G has median degree at least k, then any tree of order at most k + 1 is a subgraph of G.