763 research outputs found
Problems in extremal graph theory
We consider a variety of problems in extremal graph and set theory.
The {\em chromatic number} of , , is the smallest integer
such that is -colorable.
The {\it square} of , written , is the supergraph of in which also
vertices within distance 2 of each other in are adjacent.
A graph is a {\it minor} of if
can be obtained from a subgraph of by contracting edges.
We show that the upper bound for
conjectured by Wegner (1977) for planar graphs
holds when is a -minor-free graph.
We also show that is equal to the bound
only when contains a complete graph of that order.
One of the central problems of extremal hypergraph theory is
finding the maximum number of edges in a hypergraph
that does not contain a specific forbidden structure.
We consider as a forbidden structure a fixed number of members
that have empty common intersection
as well as small union.
We obtain a sharp upper bound on the size of uniform hypergraphs
that do not contain this structure,
when the number of vertices is sufficiently large.
Our result is strong enough to imply the same sharp upper bound
for several other interesting forbidden structures
such as the so-called strong simplices and clusters.
The {\em -dimensional hypercube}, ,
is the graph whose vertex set is and
whose edge set consists of the vertex pairs
differing in exactly one coordinate.
The generalized Tur\'an problem asks for the maximum number
of edges in a subgraph of a graph that does not contain
a forbidden subgraph .
We consider the Tur\'an problem where is and
is a cycle of length with .
Confirming a conjecture of Erd{\H o}s (1984),
we show that the ratio of the size of such a subgraph of
over the number of edges of is ,
i.e. in the limit this ratio approaches 0
as approaches infinity
Minors in expanding graphs
Extending several previous results we obtained nearly tight estimates on the
maximum size of a clique-minor in various classes of expanding graphs. These
results can be used to show that graphs without short cycles and other H-free
graphs contain large clique-minors, resolving some open questions in this area
Coloring Graphs with Forbidden Minors
Hadwiger's conjecture from 1943 states that for every integer , every
graph either can be -colored or has a subgraph that can be contracted to the
complete graph on vertices. As pointed out by Paul Seymour in his recent
survey on Hadwiger's conjecture, proving that graphs with no minor are
-colorable is the first case of Hadwiger's conjecture that is still open. It
is not known yet whether graphs with no minor are -colorable. Using a
Kempe-chain argument along with the fact that an induced path on three vertices
is dominating in a graph with independence number two, we first give a very
short and computer-free proof of a recent result of Albar and Gon\c{c}alves and
generalize it to the next step by showing that every graph with no minor
is -colorable, where . We then prove that graphs with no
minor are -colorable and graphs with no minor are
-colorable. Finally we prove that if Mader's bound for the extremal function
for minors is true, then every graph with no minor is
-colorable for all . This implies our first result. We believe
that the Kempe-chain method we have developed in this paper is of independent
interest
Constructing dense graphs with sublinear Hadwiger number
Mader asked to explicitly construct dense graphs for which the size of the
largest clique minor is sublinear in the number of vertices. Such graphs exist
as a random graph almost surely has this property. This question and variants
were popularized by Thomason over several articles. We answer these questions
by showing how to explicitly construct such graphs using blow-ups of small
graphs with this property. This leads to the study of a fractional variant of
the clique minor number, which may be of independent interest.Comment: 10 page
Disproof of the List Hadwiger Conjecture
The List Hadwiger Conjecture asserts that every -minor-free graph is
-choosable. We disprove this conjecture by constructing a
-minor-free graph that is not -choosable for every integer
Number of cliques in graphs with a forbidden subdivision
We prove that for all positive integers , every -vertex graph with no
-subdivision has at most cliques. We also prove that
asymptotically, such graphs contain at most cliques, where
tends to zero as tends to infinity. This strongly answers a question
of D. Wood asking if the number of cliques in -vertex graphs with no
-minor is at most for some constant .Comment: 10 pages; to appear in SIAM J. Discrete Mat
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