102 research outputs found
Short directed cycles in bipartite digraphs
The Caccetta-H\"aggkvist conjecture implies that for every integer ,
if is a bipartite digraph, with vertices in each part, and every vertex
has out-degree more than , then has a directed cycle of length at
most . If true this is best possible, and we prove this for
and all .
More generally, we conjecture that for every integer , and every pair
of reals with , if is a bipartite
digraph with bipartition , where every vertex in has out-degree at
least , and every vertex in has out-degree at least ,
then has a directed cycle of length at most . This implies the
Caccetta-H\"aggkvist conjecture (set and very small), and again is
best possible for infinitely many pairs . We prove this for , and prove a weaker statement (that suffices) for
Piercing axis-parallel boxes
Let \F be a finite family of axis-parallel boxes in such that \F
contains no pairwise disjoint boxes. We prove that if \F contains a
subfamily \M of pairwise disjoint boxes with the property that for every
F\in \F and M\in \M with , either contains a
corner of or contains corners of , then \F can be
pierced by points. One consequence of this result is that if and
the ratio between any of the side lengths of any box is bounded by a constant,
then \F can be pierced by points. We further show that if for each two
intersecting boxes in \F a corner of one is contained in the other, then \F
can be pierced by at most points, and in the special case
where \F contains only cubes this bound improves to
Pure pairs. I. Trees and linear anticomplete pairs
The Erdos-Hajnal Conjecture asserts that for every graph H there is a
constant c > 0 such that every graph G that does not contain H as an induced
subgraph has a clique or stable set of cardinality at least |G|^c. In this
paper, we prove a conjecture of Liebenau and Pilipczuk, that for every forest H
there exists c > 0, such that every graph G contains either an induced copy of
H, or a vertex of degree at least c|G|, or two disjoint sets of at least c|G|
vertices with no edges between them. It follows that for every forest H there
is c > 0 so that if G contains neither H nor its complement as an induced
subgraph then there is a clique or stable set of cardinality at least |G|^c
Further approximations for Aharoni's rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture
For a digraph and , let be the number of
out-neighbors of in . The Caccetta-H\"{a}ggkvist conjecture states that
for all , if is a digraph with such that for all , then contains a directed cycle of length at
most . Aharoni proposed a generalization of this conjecture,
that a simple edge-colored graph on vertices with color classes, each
of size , has a rainbow cycle of length at most . With
Pelik\'anov\'a and Pokorn\'a, we showed that this conjecture is true if each
color class has size . In this paper, we present a proof of
the conjecture if each color class has size , which improved the
previous result and is only a constant factor away from Aharoni's conjecture.
We also consider what happens when the condition on the number of colors is
relaxed
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