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 Erdős.In [2], Erdős conjectured that the number of cycles of length 5 in a triangle-free graph of order n is at most (n/5) 5 and further, this bound is attained in the case when n is divisible by 5 by the blow-up
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 P6-free graph, that is, a graph that has no path on 6 vertices as an induced subgraph. This improves the polynomial-time algorithm on P5-free graphs of Lokshtanov et al. [10], and the quasipolynomial-time algorithm on P6
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 Lovász and Sós for graphons.
In the classic
Maximum Weight Independent Set
problem, we are given a graph
G
with a nonnegative weight function on its 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. [
15
] and the quasipolynomial-time algorithm on
P
6
-free graphs of Lokshtanov et al. [
14
]. The main technical contribution leading to our main result is enumeration of a polynomial-size family ℱ of vertex subsets with the following property: For every maximal independent set
I
in the graph, ℱ contains all maximal cliques of some minimal chordal completion of
G
that does not add any edge incident to a vertex of
I
.
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