458,745 research outputs found
Uniqueness and minimal obstructions for tree-depth
A k-ranking of a graph G is a labeling of the vertices of G with values from
{1,...,k} such that any path joining two vertices with the same label contains
a vertex having a higher label. The tree-depth of G is the smallest value of k
for which a k-ranking of G exists. The graph G is k-critical if it has
tree-depth k and every proper minor of G has smaller tree-depth.
We establish partial results in support of two conjectures about the order
and maximum degree of k-critical graphs. As part of these results, we define a
graph G to be 1-unique if for every vertex v in G, there exists an optimal
ranking of G in which v is the unique vertex with label 1. We show that several
classes of k-critical graphs are 1-unique, and we conjecture that the property
holds for all k-critical graphs. Generalizing a previously known construction
for trees, we exhibit an inductive construction that uses 1-unique k-critical
graphs to generate large classes of critical graphs having a given tree-depth.Comment: 14 pages, 4 figure
Exhaustive generation of -critical -free graphs
We describe an algorithm for generating all -critical -free
graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove
that there are only finitely many -critical -free graphs, for
both and . We also show that there are only finitely many
-critical graphs -free graphs. For each case of these cases we
also give the complete lists of critical graphs and vertex-critical graphs.
These results generalize previous work by Hell and Huang, and yield certifying
algorithms for the -colorability problem in the respective classes.
Moreover, we prove that for every , the class of 4-critical planar
-free graphs is finite. We also determine all 27 4-critical planar
-free graphs.
We also prove that every -free graph of girth at least five is
3-colorable, and determine the smallest 4-chromatic -free graph of
girth five. Moreover, we show that every -free graph of girth at least
six and every -free graph of girth at least seven is 3-colorable. This
strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with
arXiv:1504.0697
Riemann-Roch and Abel-Jacobi theory on a finite graph
It is well-known that a finite graph can be viewed, in many respects, as a
discrete analogue of a Riemann surface. In this paper, we pursue this analogy
further in the context of linear equivalence of divisors. In particular, we
formulate and prove a graph-theoretic analogue of the classical Riemann-Roch
theorem. We also prove several results, analogous to classical facts about
Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian.
As an application of our results, we characterize the existence or
non-existence of a winning strategy for a certain chip-firing game played on
the vertices of a graph.Comment: 35 pages. v3: Several minor changes made, mostly fixing typographical
errors. This is the final version, to appear in Adv. Mat
Constant mean curvature surfaces in 3-dimensional Thurston geometries
This is a survey on the global theory of constant mean curvature surfaces in
Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight
canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 \times R, S2
\times R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie
group Sol3. We will focus on the problems of classifying compact CMC surfaces
and entire CMC graphs in these spaces. A collection of important open problems
of the theory is also presented
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