35 research outputs found
The Erd\H{o}s-Hajnal conjecture for caterpillars and their complements
The celebrated Erd\H{o}s-Hajnal conjecture states that for every proper
hereditary graph class there exists a constant such that every graph
contains a clique or an independent set of size . Recently,
there has been a growing interest in the symmetrized variant of this
conjecture, where one additionally requires to be closed under
complementation.
We show that any hereditary graph class that is closed under complementation
and excludes a fixed caterpillar as an induced subgraph satisfies the
Erd\H{o}s-Hajnal conjecture. Here, a caterpillar is a tree whose vertices of
degree at least three lie on a single path (i.e., our caterpillars may have
arbitrarily long legs). In fact, we prove a stronger property of such graph
classes, called in the literature the strong Erd\H{o}s-Hajnal property: for
every such graph class , there exists a constant such that every graph contains two
disjoint sets of size at least each so that
either all edges between and are present in , or none of them. This
result significantly extends the family of graph classes for which we know that
the strong Erd\H{o}s-Hajnal property holds; for graph classes excluding a graph
and its complement it was previously known only for paths [Bousquet,
Lagoutte, Thomass\'{e}, JCTB 2015] and hooks (i.e., paths with an additional
pendant vertex at third vertex of the path) [Choromanski, Falik, Liebenau,
Patel, Pilipczuk, arXiv:1508.00634]
Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph
In this paper we relate a fundamental parameter of a random graph, its degree
sequence, to a simple model of nearly independent binomial random variables.
This confirms a conjecture made in 1997. As a result, many interesting
functions of the joint distribution of graph degrees, such as the distribution
of the median degree, become amenable to estimation. Our result is established
by proving an asymptotic formula conjectured in 1990 for the number of graphs
with given degree sequence. In particular, this gives an asymptotic formula for
the number of -regular graphs for all , as .Comment: The new version contains a modification of the main proof that makes
the calculation component easie
A non-trivial upper bound on the threshold bias of the Oriented-cycle game
In the Oriented-cycle game, introduced by Bollob\'as and Szab\'o, two
players, called OMaker and OBreaker, alternately direct edges of . OMaker
directs exactly one edge, whereas OBreaker is allowed to direct between one and
edges. OMaker wins if the final tournament contains a directed cycle,
otherwise OBreaker wins. Bollob\'as and Szab\'o conjectured that for a bias as
large as OMaker has a winning strategy if OBreaker must take exactly
edges in each round. It was shown recently by Ben-Eliezer, Krivelevich and
Sudakov, that OMaker has a winning strategy for this game whenever . In this paper, we show that OBreaker has a winning strategy
whenever . Moreover, in case OBreaker is required to
direct exactly edges in each move, we show that OBreaker wins for , provided that is large enough. This refutes the conjecture
by Bollob\'as and Szab\'o.Comment: 31 pages, 7 figure
Ramsey equivalence of and
We prove that, for , the graphs and are Ramsey
equivalent. That is, if is such that any red-blue colouring of its edges
creates a monochromatic then it must also possess a monochromatic
On the Concentration of the Domination Number of the Random Graph
In this paper we study the behaviour of the domination number of the
Erd\H{o}s-R\'enyi random graph . Extending a result of
Wieland and Godbole we show that the domination number of is
equal to one of two values asymptotically almost surely whenever . The explicit values are exactly at the first moment
threshold, that is where the expected number of dominating sets starts to tend
to infinity. For small we also provide various non-concentration results
which indicate why some sort of lower bound on the probability is necessary
in our first theorem. Concentration, though not on a constant length interval,
is proven for every . These results show that unlike in the case of
where concentration of the domination number
happens around the first moment threshold, for it does so
around the median. In particular, in this range the two are far apart from each
other
The random graph intuition for the tournament game
In the tournament game two players, called Maker and Breaker, alternately
take turns in claiming an unclaimed edge of the complete graph on n vertices
and selecting one of the two possible orientations. Before the game starts,
Breaker fixes an arbitrary tournament T_k on k vertices. Maker wins if, at the
end of the game, her digraph contains a copy of T_k; otherwise Breaker wins. In
our main result, we show that Maker has a winning strategy for k =
(2-o(1))log_2 n, improving the constant factor in previous results of Beck and
the second author. This is asymptotically tight since it is known that for k =
(2-o(1))log_2 n Breaker can prevent that the underlying graph of Maker's graph
contains a k-clique. Moreover the precise value of our lower bound differs from
the upper bound only by an additive constant of 12.
We also discuss the question whether the random graph intuition, which
suggests that the threshold for k is asymptotically the same for the game
played by two "clever" players and the game played by two "random" players, is
supported by the tournament game: It will turn out that, while a
straightforward application of this intuition fails, a more subtle version of
it is still valid.
Finally, we consider the orientation-game version of the tournament game,
where Maker wins the game if the final digraph -- containing also the edges
directed by Breaker -- possesses a copy of T_k. We prove that in that game
Breaker has a winning strategy for k = (4+o(1))log_2 n
On minimal Ramsey graphs and Ramsey equivalence in multiple colours
For an integer , a graph is called -Ramsey for a graph if
every -colouring of the edges of contains a monochromatic copy of .
If is -Ramsey for , yet no proper subgraph of has this property
then is called -Ramsey-minimal for . Generalising a statement by
Burr, Ne\v{s}et\v{r}il and R\"odl from 1977 we prove that, for , if
is a graph that is not -Ramsey for some graph then is contained as
an induced subgraph in an infinite number of -Ramsey-minimal graphs for ,
as long as is -connected or isomorphic to the triangle. For such ,
the following are some consequences.
(1) For , every -Ramsey-minimal graph for is contained as
an induced subgraph in an infinite number of -Ramsey-minimal graphs for .
(2) For every , there are -Ramsey-minimal graphs for of
arbitrarily large maximum degree, genus, and chromatic number.
(3) The collection
forms an antichain with respect to the subset relation, where
denotes the set of all graphs that are -Ramsey-minimal for .
We also address the question which pairs of graphs satisfy , in which case and are called
-equivalent. We show that two graphs and are -equivalent for
even if they are -equivalent, and that in general -equivalence for
some does not necessarily imply -equivalence. Finally we indicate
that for connected graphs this implication may hold: Results by
Ne\v{s}et\v{r}il and R\"odl and by Fox, Grinshpun, Liebenau, Person and Szab\'o
imply that the complete graph is not -equivalent to any other connected
graph. We prove that this is the case for an arbitrary number of colours
Fast strategies in Maker-Breaker games played on random boards
In this paper we analyze classical Maker-Breaker games played on the edge set
of a sparse random board G\sim \gnp. We consider the Hamiltonicity game, the
perfect matching game and the -connectivity game. We prove that for
, the board G\sim \gnp is typically such that
Maker can win these games asymptotically as fast as possible, i.e. within
, and moves respectively
Asymptotic enumeration of digraphs and bipartite graphs by degree sequence
We provide asymptotic formulae for the numbers of bipartite graphs with given
degree sequence, and of loopless digraphs with given in- and out-degree
sequences, for a wide range of parameters. Our results cover medium range
densities and close the gaps between the results known for the sparse and dense
ranges. In the case of bipartite graphs, these results were proved by
Greenhill, McKay and Wang in 2006 and by Canfield, Greenhill and McKay in 2008,
respectively. Our method also essentially covers the sparse range, for which
much less was known in the case of loopless digraphs. For the range of
densities which our results cover, they imply that the degree sequence of a
random bipartite graph with m edges is accurately modelled by a sequence of
independent binomial random variables, conditional upon the sum of variables in
each part being equal to m. A similar model also holds for loopless digraphs
Caterpillars in Erd\H{o}s-Hajnal
Let be a tree such that all its vertices of degree more than two lie on
one path, that is, is a caterpillar subdivision. We prove that there exists
such that for every graph with not containing
as an induced subgraph, either some vertex has at least
neighbours, or there are two disjoint sets of vertices , both of
cardinality at least , where there is no edge joining and
.
A consequence is: for every caterpillar subdivision , there exists
such that for every graph containing neither of and its complement as
an induced subgraph, has a clique or stable set with at least
vertices. This extends a theorem of Bousquet, Lagoutte and Thomass\'e [JCTB
2015], who proved the same when is a path, and a recent theorem of
Choromanski, Falik, Liebenau, Patel and Pilipczuk [Electron. J. Combin. 2018],
who proved it when is a "hook"