270 research outputs found
Class two 1-planar graphs with maximum degree six or seven
A graph is 1-planar if it can be drawn on the plane so that each edge is
crossed by at most one other edge. In this note we give examples of class two
1-planar graphs with maximum degree six or seven.Comment: 3 pages, 2 figure
Distributed Algorithms, the Lov\'{a}sz Local Lemma, and Descriptive Combinatorics
In this paper we consider coloring problems on graphs and other combinatorial
structures on standard Borel spaces. Our goal is to obtain sufficient
conditions under which such colorings can be made well-behaved in the sense of
topology or measure. To this end, we show that such well-behaved colorings can
be produced using certain powerful techniques from finite combinatorics and
computer science. First, we prove that efficient distributed coloring
algorithms (on finite graphs) yield well-behaved colorings of Borel graphs of
bounded degree; roughly speaking, deterministic algorithms produce Borel
colorings, while randomized algorithms give measurable and Baire-measurable
colorings. Second, we establish measurable and Baire-measurable versions of the
Symmetric Lov\'{a}sz Local Lemma (under the assumption
, which is stronger than the standard
LLL assumption but still sufficient
for many applications). From these general results, we derive a number of
consequences in descriptive combinatorics and ergodic theory.Comment: 35 page
Boxicity of graphs on surfaces
The boxicity of a graph is the least integer for which there
exist interval graphs , , such that . Scheinerman proved in 1984 that outerplanar graphs have boxicity
at most two and Thomassen proved in 1986 that planar graphs have boxicity at
most three. In this note we prove that the boxicity of toroidal graphs is at
most 7, and that the boxicity of graphs embeddable in a surface of
genus is at most . This result yields improved bounds on the
dimension of the adjacency poset of graphs on surfaces.Comment: 9 pages, 2 figure
Colorings of oriented planar graphs avoiding a monochromatic subgraph
For a fixed simple digraph and a given simple digraph , an -free
-coloring of is a vertex-coloring in which no induced copy of in
is monochromatic. We study the complexity of deciding for fixed and
whether a given simple digraph admits an -free -coloring. Our main focus
is on the restriction of the problem to planar input digraphs, where it is only
interesting to study the cases . From known results it follows
that for every fixed digraph whose underlying graph is not a forest, every
planar digraph admits an -free -coloring, and that for every fixed
digraph with , every oriented planar graph admits an
-free -coloring.
We show in contrast, that
- if is an orientation of a path of length at least , then it is
NP-hard to decide whether an acyclic and planar input digraph admits an
-free -coloring.
- if is an orientation of a path of length at least , then it is
NP-hard to decide whether an acyclic and planar input digraph admits an
-free -coloring
- …