On the maximum number of five-cycles in a triangle-free graph

Using Razborov's flag algebras we show that a triangle-free graph on n vertices contains at most (n/5)^5 cycles of length five. It settles in the affirmative a conjecture of Erdos.Comment: After minor revisions; to appear in JCT

Elusive extremal graphs

We study the uniqueness of optimal solutions to extremal graph theory problems. Lovasz conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the resulting set is satisfied by an asymptotically unique graph. This statement is often referred to as saying that `every extremal graph theory problem has a finitely forcible optimum'. We present a counterexample to the conjecture. Our techniques also extend to a more general setting involving other types of constraints

Polynomial-time algorithm for Maximum Weight Independent Set on P6P_6-free graphs

In the classic Maximum Weight Independent Set problem we are given a graph $G$ with a nonnegative weight function on vertices, and the goal is to find an independent set in $G$ of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any $P_6$-free graph, that is, a graph that has no path on $6$ vertices as an induced subgraph. This improves the polynomial-time algorithm on $P_5$-free graphs of Lokshtanov et al. (SODA 2014), and the quasipolynomial-time algorithm on $P_6$-free graphs of Lokshtanov et al (SODA 2016). The main technical contribution leading to our main result is enumeration of a polynomial-size family $\mathcal{F}$ of vertex subsets with the following property: for every maximal independent set $I$ in the graph, $\mathcal{F}$ contains all maximal cliques of some minimal chordal completion of $G$ that does not add any edge incident to a vertex of $I$

Finitely forcible graphons and permutons

We investigate when limits of graphs (graphons) and permutations (permutons) are uniquely determined by finitely many densities of their substructures, i.e., when they are finitely forcible. Every permuton can be associated with a graphon through the notion of permutation graphs. We find permutons that are finitely forcible but the associated graphons are not. We also show that all permutons that can be expressed as a finite combination of monotone permutons and quasirandom permutons are finitely forcible, which is the permuton counterpart of the result of Lovasz and Sos for graphons.Comment: 30 pages, 18 figure

On the maximum number of odd cycles in graphs without smaller odd cycles

We prove that for each odd integer $k \geq 7$, every graph on $n$ vertices without odd cycles of length less than $k$ contains at most $(n/k)^k$ cycles of length $k$. This generalizes the previous results on the maximum number of pentagons in triangle-free graphs, conjectured by Erd\H{o}s in 1984, and asymptotically determines the generalized Tur\'an number $\mathrm{ex}(n,C_k,C_{k-2})$ for odd $k$. In contrary to the previous results on the pentagon case, our proof is not computer-assisted

Graphs without a rainbow path of length 3

In 1959 Erd\H{o}s and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. Here we study a rainbow version of their theorem, in which one considers $k \geq 1$ graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximal number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any $k \geq 1$

Cycles of length three and four in tournaments

Linial and Morgenstern conjectured that, among all $n$-vertex tournaments with $d\binom{n}{3}$ cycles of length three, the number of cycles of length four is asymptotically minimized by a random blow-up of a transitive tournament with all but one part of equal size and one smaller part. We prove the conjecture for $d\ge 1/36$ by analyzing the possible spectrum of adjacency matrices of tournaments. We also demonstrate that the family of extremal examples is broader than expected and give its full description for $d\ge 1/16$