23 research outputs found
Defective and Clustered Choosability of Sparse Graphs
An (improper) graph colouring has "defect" if each monochromatic subgraph
has maximum degree at most , and has "clustering" if each monochromatic
component has at most vertices. This paper studies defective and clustered
list-colourings for graphs with given maximum average degree. We prove that
every graph with maximum average degree less than is
-choosable with defect . This improves upon a similar result by Havet and
Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with
maximum average degree , no bound on the number of colours
was previously known. The above result with solves this problem. It
implies that every graph with maximum average degree is
-choosable with clustering 2. This extends a
result of Kopreski and Yu [Discrete Math., 2017] to the setting of
choosability. We then prove two results about clustered choosability that
explore the trade-off between the number of colours and the clustering. In
particular, we prove that every graph with maximum average degree is
-choosable with clustering , and is
-choosable with clustering . As an
example, the later result implies that every biplanar graph is 8-choosable with
bounded clustering. This is the best known result for the clustered version of
the earth-moon problem. The results extend to the setting where we only
consider the maximum average degree of subgraphs with at least some number of
vertices. Several applications are presented
Rainbow saturation for complete graphs
We call an edge-colored graph rainbow if all of its edges receive distinct
colors. An edge-colored graph is called -rainbow saturated if
does not contain a rainbow copy of and adding an edge of any color
to creates a rainbow copy of . The rainbow saturation number
is the minimum number of edges in an -vertex -rainbow
saturated graph. Gir\~{a}o, Lewis, and Popielarz conjectured that
for fixed . Disproving this conjecture,
we establish that for every , there exists a constant such
that Recently, Behague,
Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker
upper bound which was sufficient to disprove the conjecture. They also
introduced the weak rainbow saturation number, and asked whether this is equal
to the rainbow saturation number of , since the standard weak saturation
number of complete graphs equals the standard saturation number. Surprisingly,
our lower bound separates the rainbow saturation number from the weak rainbow
saturation number, answering this question in the negative. The existence of
the constant resolves another of their questions in the affirmative
for complete graphs. Furthermore, we show that the conjecture of Gir\~{a}o,
Lewis, and Popielarz is true if we have an additional assumption that the
edge-colored -rainbow saturated graph must be rainbow. As an ingredient of
the proof, we study graphs which are -saturated with respect to the
operation of deleting one edge and adding two edges
On an induced version of Menger's theorem
We prove Menger-type results in which the obtained paths are pairwise
non-adjacent, both for graphs of bounded maximum degree and, more generally,
for graphs excluding a topological minor. We further show better bounds in the
subcubic case, and in particular obtain a tight result for two paths using a
computer-assisted proof.Comment: 14 pages, 4 figure
Obstructions for bounded branch-depth in matroids
DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a
natural analogue of tree-depth of graphs. They conjectured that a matroid of
sufficiently large branch-depth contains the uniform matroid or the
cycle matroid of a large fan graph as a minor. We prove that matroids with
sufficiently large branch-depth either contain the cycle matroid of a large fan
graph as a minor or have large branch-width. As a corollary, we prove their
conjecture for matroids representable over a fixed finite field and
quasi-graphic matroids, where the uniform matroid is not an option.Comment: 25 pages, 1 figur
A unified Erd\H{o}s-P\'{o}sa theorem for cycles in graphs labelled by multiple abelian groups
In 1965, Erd\H{o}s and P\'{o}sa proved that there is a duality between the
maximum size of a packing of cycles and the minimum size of a vertex set
hitting all cycles. Such a duality does not hold for odd cycles, and Dejter and
Neumann-Lara asked in 1988 to find all pairs of integers where
such a duality holds for the family of cycles of length modulo . We
characterise all such pairs, and we further generalise this characterisation to
cycles in graphs labelled with a bounded number of abelian groups, whose values
avoid a bounded number of elements of each group. This unifies almost all known
types of cycles that admit such a duality, and it also provides new results.
Moreover, we characterise the obstructions to such a duality in this setting,
and thereby obtain an analogous characterisation for cycles in graphs
embeddable on a fixed compact orientable surface.Comment: 37 pages, 2 figure
Product structure of graph classes with bounded treewidth
We show that many graphs with bounded treewidth can be described as subgraphs
of the strong product of a graph with smaller treewidth and a bounded-size
complete graph. To this end, define the "underlying treewidth" of a graph class
to be the minimum non-negative integer such that, for some
function , for every graph there is a graph with
such that is isomorphic to a subgraph of . We introduce disjointed coverings of graphs
and show they determine the underlying treewidth of any graph class. Using this
result, we prove that the class of planar graphs has underlying treewidth 3;
the class of -minor-free graphs has underlying treewidth (for ); and the class of -minor-free graphs has underlying
treewidth . In general, we prove that a monotone class has bounded
underlying treewidth if and only if it excludes some fixed topological minor.
We also study the underlying treewidth of graph classes defined by an excluded
subgraph or excluded induced subgraph. We show that the class of graphs with no
subgraph has bounded underlying treewidth if and only if every component of
is a subdivided star, and that the class of graphs with no induced
subgraph has bounded underlying treewidth if and only if every component of
is a star
Tropospheric O 3 moderates responses of temperate hardwood forests to elevated CO 2 : a synthesis of molecular to ecosystem results from the Aspen FACE project
1. The impacts of elevated atmospheric CO 2 and/or O 3 have been examined over 4 years using an open-air exposure system in an aggrading northern temperate forest containing two different functional groups (the indeterminate, pioneer, O 3 -sensitive species Trembling Aspen, Populus tremuloides and Paper Birch, Betula papyrifera , and the determinate, late successional, O 3 -tolerant species Sugar Maple, Acer saccharum ). 2. The responses to these interacting greenhouse gases have been remarkably consistent in pure Aspen stands and in mixed Aspen/Birch and Aspen/Maple stands, from leaf to ecosystem level, for O 3 -tolerant as well as O 3 -sensitive genotypes and across various trophic levels. These two gases act in opposing ways, and even at low concentrations (1·5 × ambient, with ambient averaging 34–36 nL L −1 during the summer daylight hours), O 3 offsets or moderates the responses induced by elevated CO 2 . 3. After 3 years of exposure to 560 µmol mol −1 CO 2 , the above-ground volume of Aspen stands was 40% above those grown at ambient CO 2 , and there was no indication of a diminishing growth trend. In contrast, O 3 at 1·5 × ambient completely offset the growth enhancement by CO 2 , both for O 3 -sensitive and O 3 -tolerant clones. Implications of this finding for carbon sequestration, plantations to reduce excess CO 2 , and global models of forest productivity and climate change are presented.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/72125/1/j.1365-2435.2003.00733.x.pd