24 research outputs found
The Erd\H{o}s-Hajnal Conjecture for Paths and Antipaths
We prove that for every k, there exists such that every graph G on n
vertices not inducing a path and its complement contains a clique or a
stable set of size
Excluding pairs of tournaments
The Erd\H{o}s-Hajnal conjecture states that for every given undirected graph
there exists a constant such that every graph that does not
contain as an induced subgraph contains a clique or a stable set of size at
least . The conjecture is still open. Its equivalent directed
version states that for every given tournament there exists a constant
such that every -free tournament contains a transitive
subtournament of order at least . We prove in this paper that
-free tournaments contain transitive subtournaments of
size at least for some and several
pairs of tournaments: , . In particular we prove that
-freeness implies existence of the polynomial-size transitive
subtournaments for several tournaments for which the conjecture is still
open ( stands for the \textit{complement of }). To the best of our
knowledge these are first nontrivial results of this type
Clique versus Independent Set
Yannakakis' Clique versus Independent Set problem (CL-IS) in communication
complexity asks for the minimum number of cuts separating cliques from stable
sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial
CS-separator, i.e. of size , and addresses the problem of
finding a polynomial CS-separator. This question is still open even for perfect
graphs. We show that a polynomial CS-separator almost surely exists for random
graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a
clique and a stable set) then there exists a constant for which we find a
CS-separator on the class of H-free graphs. This generalizes a
result of Yannakakis on comparability graphs. We also provide a
CS-separator on the class of graphs without induced path of length k and its
complement. Observe that on one side, is of order
resulting from Vapnik-Chervonenkis dimension, and on the other side, is
exponential.
One of the main reason why Yannakakis' CL-IS problem is fascinating is that
it admits equivalent formulations. Our main result in this respect is to show
that a polynomial CS-separator is equivalent to the polynomial
Alon-Saks-Seymour Conjecture, asserting that if a graph has an edge-partition
into k complete bipartite graphs, then its chromatic number is polynomially
bounded in terms of k. We also show that the classical approach to the stubborn
problem (arising in CSP) which consists in covering the set of all solutions by
instances of 2-SAT is again equivalent to the existence of a
polynomial CS-separator
Towards the Erd\H{o}s-Hajnal conjecture for -free graphs
The Erd\H{o}s-Hajnal conjecture is one of the most classical and well-known
problems in extremal and structural combinatorics dating back to 1977. It
asserts that in stark contrast to the case of a general -vertex graph if one
imposes even a little bit of structure on the graph, namely by forbidding a
fixed graph as an induced subgraph, instead of only being able to find a
polylogarithmic size clique or an independent set one can find one of
polynomial size. Despite being the focus of considerable attention over the
years the conjecture remains open. In this paper we improve the best known
lower bound of on this question, due to Erd\H{o}s
and Hajnal from 1989, in the smallest open case, namely when one forbids a
, the path on vertices. Namely, we show that any -free vertex
graph contains a clique or an independent set of size at least . Our methods also lead to the same improvement for an infinite
family of graphs