126 research outputs found
Note on the number of edges in families with linear union-complexity
We give a simple argument showing that the number of edges in the
intersection graph of a family of sets in the plane with a linear
union-complexity is . In particular, we prove for intersection graph of a family of
pseudo-discs, which improves a previous bound.Comment: background and related work is now more complete; presentation
improve
An on-line competitive algorithm for coloring bipartite graphs without long induced paths
The existence of an on-line competitive algorithm for coloring bipartite
graphs remains a tantalizing open problem. So far there are only partial
positive results for bipartite graphs with certain small forbidden graphs as
induced subgraphs. We propose a new on-line competitive coloring algorithm for
-free bipartite graphs
Planar posets have dimension at most linear in their height
We prove that every planar poset of height has dimension at most
. This improves on previous exponential bounds and is best possible
up to a constant factor. We complement this result with a construction of
planar posets of height and dimension at least .Comment: v2: Minor change
Towards on-line Ohba's conjecture
The on-line choice number of a graph is a variation of the choice number
defined through a two person game. It is at least as large as the choice number
for all graphs and is strictly larger for some graphs. In particular, there are
graphs with whose on-line choice numbers are larger
than their chromatic numbers, in contrast to a recently confirmed conjecture of
Ohba that every graph with has its choice number
equal its chromatic number. Nevertheless, an on-line version of Ohba conjecture
was proposed in [P. Huang, T. Wong and X. Zhu, Application of polynomial method
to on-line colouring of graphs, European J. Combin., 2011]: Every graph
with has its on-line choice number equal its chromatic
number. This paper confirms the on-line version of Ohba conjecture for graphs
with independence number at most 3. We also study list colouring of
complete multipartite graphs with all parts of size 3. We prove
that the on-line choice number of is at most , and
present an alternate proof of Kierstead's result that its choice number is
. For general graphs , we prove that if then its on-line choice number equals chromatic number.Comment: new abstract and introductio
Boolean dimension and tree-width
The dimension is a key measure of complexity of partially ordered sets. Small
dimension allows succinct encoding. Indeed if has dimension , then to
know whether in it is enough to check whether in each
of the linear extensions of a witnessing realizer. Focusing on the encoding
aspect Ne\v{s}et\v{r}il and Pudl\'{a}k defined a more expressive version of
dimension. A poset has boolean dimension at most if it is possible to
decide whether in by looking at the relative position of and
in only permutations of the elements of . We prove that posets with
cover graphs of bounded tree-width have bounded boolean dimension. This stays
in contrast with the fact that there are posets with cover graphs of tree-width
three and arbitrarily large dimension. This result might be a step towards a
resolution of the long-standing open problem: Do planar posets have bounded
boolean dimension?Comment: one more reference added; paper revised along the suggestion of three
reviewer
Boolean Dimension, Components and Blocks
We investigate the behavior of Boolean dimension with respect to components
and blocks. To put our results in context, we note that for Dushnik-Miller
dimension, we have that if for every component of a poset
, then ; also if for every block
of a poset , then . By way of constrast, local dimension is
well behaved with respect to components, but not for blocks: if
for every component of a poset , then
; however, for every , there exists a poset
with and for every block of . In this
paper we show that Boolean dimension behaves like Dushnik-Miller dimension with
respect to both components and blocks: if for every
component of , then ; also if
for every block of , then .Comment: 12 pages. arXiv admin note: text overlap with arXiv:1712.0609
A note on concurrent graph sharing games
In the concurrent graph sharing game, two players, called First and Second,
share the vertices of a connected graph with positive vertex-weights summing up
to as follows. The game begins with First taking any vertex. In each
proceeding round, the player with the smaller sum of collected weights so far
chooses a non-taken vertex adjacent to a vertex which has been taken, i.e., the
set of all taken vertices remains connected and one new vertex is taken in
every round. (It is assumed that no two subsets of vertices have the same sum
of weights.) One can imagine the players consume their taken vertex over a time
proportional to its weight, before choosing a next vertex. In this note we show
that First has a strategy to guarantee vertices of weight at least
regardless of the graph and how it is weighted. This is best-possible already
when the graph is a cycle. Moreover, if the graph is a tree First can guarantee
vertices of weight at least , which is clearly best-possible.Comment: expanded introduction and conclusion
Pathwidth and nonrepetitive list coloring
A vertex coloring of a graph is nonrepetitive if there is no path in the
graph whose first half receives the same sequence of colors as the second half.
While every tree can be nonrepetitively colored with a bounded number of colors
(4 colors is enough), Fiorenzi, Ochem, Ossona de Mendez, and Zhu recently
showed that this does not extend to the list version of the problem, that is,
for every there is a tree that is not nonrepetitively
-choosable. In this paper we prove the following positive result, which
complements the result of Fiorenzi et al.: There exists a function such
that every tree of pathwidth is nonrepetitively -choosable. We also
show that such a property is specific to trees by constructing a family of
pathwidth-2 graphs that are not nonrepetitively -choosable for any fixed
.Comment: v2: Minor changes made following helpful comments by the referee
Coloring intersection graphs of arc-connected sets in the plane
A family of sets in the plane is simple if the intersection of its any
subfamily is arc-connected, and it is pierced by a line if the intersection
of its any member with is a nonempty segment. It is proved that the
intersection graphs of simple families of compact arc-connected sets in the
plane pierced by a common line have chromatic number bounded by a function of
their clique number.Comment: Minor changes + some additional references not included in the
journal versio
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