144 research outputs found
On the odd girth and the circular chromatic number of generalized Petersen graphs
A class of simple graphs such as is said to be {\it
odd-girth-closed} if for any positive integer there exists a graph such that the odd-girth of is greater than or equal to . An
odd-girth-closed class of graphs is said to be {\it odd-pentagonal}
if there exists a positive integer depending on such that any
graph whose odd-girth is greater than admits a
homomorphism to the five cycle (i.e. is -colorable).
In this article, we show that finding the odd girth of generalized Petersen
graphs can be transformed to an integer programming problem, and using this we
explicitly compute the odd girth of such graphs, showing that the class is
odd-girth-closed. Also, motivated by showing that the class of generalized
Petersen graphs is odd-pentagonal, we study the circular chromatic number of
such graphs
On tension-continuous mapings
Tension-continuous (shortly TT) mappings are mappings between the edge sets
of graphs. They generalize graph homomorphisms. From another perspective,
tension-continuous mappings are dual to the notion of flow-continuous mappings
and the context of nowhere-zero flows motivates several questions considered in
this paper.
Extending our earlier research we define new constructions and operations for
graphs (such as graphs Delta(G)) and give evidence for the complex relationship
of homomorphisms and TT mappings. Particularly, solving an open problem, we
display pairs of TT-comparable and homomorphism-incomparable graphs with
arbitrarily high connectivity.
We give a new (and more direct) proof of density of TT order and study graphs
such that TT mappings and homomorphisms from them coincide; we call such graphs
homotens. We show that most graphs are homotens, on the other hand every vertex
of a nontrivial homotens graph is contained in a triangle. This provides a
justification for our construction of homotens graphs.Comment: 31 page
Optimal induced universal graphs for bounded-degree graphs
We show that for any constant , there exists a graph with
vertices which contains every -vertex graph with maximum
degree as an induced subgraph. For odd this significantly
improves the best-known earlier bound of Esperet et al. and is optimal up to a
constant factor, as it is known that any such graph must have at least
vertices.
Our proof builds on the approach of Alon and Capalbo (SODA 2008) together
with several additional ingredients. The construction of is explicit and is
based on an appropriately defined composition of high-girth expander graphs.
The proof also provides an efficient deterministic procedure for finding, for
any given input graph on vertices with maximum degree at most ,
an induced subgraph of isomorphic to
Approximability Distance in the Space of H-Colourability Problems
A graph homomorphism is a vertex map which carries edges from a source graph
to edges in a target graph. We study the approximability properties of the
Weighted Maximum H-Colourable Subgraph problem (MAX H-COL). The instances of
this problem are edge-weighted graphs G and the objective is to find a subgraph
of G that has maximal total edge weight, under the condition that the subgraph
has a homomorphism to H; note that for H=K_k this problem is equivalent to MAX
k-CUT. To this end, we introduce a metric structure on the space of graphs
which allows us to extend previously known approximability results to larger
classes of graphs. Specifically, the approximation algorithms for MAX CUT by
Goemans and Williamson and MAX k-CUT by Frieze and Jerrum can be used to yield
non-trivial approximation results for MAX H-COL. For a variety of graphs, we
show near-optimality results under the Unique Games Conjecture. We also use our
method for comparing the performance of Frieze & Jerrum's algorithm with
Hastad's approximation algorithm for general MAX 2-CSP. This comparison is, in
most cases, favourable to Frieze & Jerrum.Comment: 19 page
Classification on Large Networks: A Quantitative Bound via Motifs and Graphons
When each data point is a large graph, graph statistics such as densities of
certain subgraphs (motifs) can be used as feature vectors for machine learning.
While intuitive, motif counts are expensive to compute and difficult to work
with theoretically. Via graphon theory, we give an explicit quantitative bound
for the ability of motif homomorphisms to distinguish large networks under both
generative and sampling noise. Furthermore, we give similar bounds for the
graph spectrum and connect it to homomorphism densities of cycles. This results
in an easily computable classifier on graph data with theoretical performance
guarantee. Our method yields competitive results on classification tasks for
the autoimmune disease Lupus Erythematosus.Comment: 17 pages, 2 figures, 1 tabl
Tension continuous maps--their structure and applications
We consider mappings between edge sets of graphs that lift tensions to
tensions. Such mappings are called tension-continuous mappings (shortly TT
mappings). Existence of a TT mapping induces a (quasi)order on the class of
graphs, which seems to be an essential extension of the homomorphism order
(studied extensively, see [Hell-Nesetril]). In this paper we study the
relationship of the homomorphism and TT orders. We stress the similarities and
the differences in both deterministic and random setting. Particularly, we
prove that TT order is dense and universal and we solve a problem of M. DeVos
et al.Comment: 32 page
On the Impact of Dimension Reduction on Graphical Structures
Statisticians and quantitative neuroscientists have actively promoted the use
of independence relationships for investigating brain networks, genomic
networks, and other measurement technologies. Estimation of these graphs
depends on two steps. First is a feature extraction by summarizing measurements
within a parcellation, regional or set definition to create nodes. Secondly,
these summaries are then used to create a graph representing relationships of
interest. In this manuscript we study the impact of dimension reduction on
graphs that describe different notions of relations among a set of random
variables. We are particularly interested in undirected graphs that capture the
random variables' independence and conditional independence relations. A
dimension reduction procedure can be any mapping from high dimensional spaces
to low dimensional spaces. We exploit a general framework for modeling the raw
data and advocate that in estimating the undirected graphs, any acceptable
dimension reduction procedure should be a graph-homotopic mapping, i.e., the
graphical structure of the data after dimension reduction should inherit the
main characteristics of the graphical structure of the raw data. We show that,
in terms of inferring undirected graphs that characterize the conditional
independence relations among random variables, many dimension reduction
procedures, such as the mean, median, or principal components, cannot be
theoretically guaranteed to be a graph-homotopic mapping. The implications of
this work are broad. In the most charitable setting for researchers, where the
correct node definition is known, graphical relationships can be contaminated
merely via the dimension reduction. The manuscript ends with a concrete
example, characterizing a subset of graphical structures such that the
dimension reduction procedure using the principal components can be a
graph-homotopic mapping.Comment: 19 page
Homomorphism counts in robustly sparse graphs
For a fixed graph and for arbitrarily large host graphs , the number
of homomorphisms from to and the number of subgraphs isomorphic to
contained in have been extensively studied in extremal graph theory and
graph limits theory when the host graphs are allowed to be dense. This paper
addresses the case when the host graphs are robustly sparse and proves a
general theorem that solves a number of open questions proposed since 1990s and
strengthens a number of results in the literature.
We prove that for any graph and any set of homomorphisms
from to members of a hereditary class of graphs, if
satisfies a natural and mild condition, and contracting disjoint
subgraphs of radius in members of cannot
create a graph with large edge-density, then an obvious lower bound for the
size of gives a good estimation for the size of .
This result determines the maximum number of -homomorphisms, the maximum
number of -subgraphs, and the maximum number -induced subgraphs in graphs
in any hereditary class with bounded expansion up to a constant factor; it also
determines the exact value of the asymptotic logarithmic density for
-homomorphisms, -subgraphs and -induced subgraphs in graphs in any
hereditary nowhere dense class. Hereditary classes with bounded expansion
include (topological) minor-closed families and many classes of graphs with
certain geometric properties; nowhere dense classes are the most general sparse
classes in sparsity theory. Our machinery also allows us to determine the
maximum number of -subgraphs in the class of all -degenerate graphs with
any fixed
Coloring dense graphs via VC-dimension
The Vapnik-\v{C}ervonenkis dimension is a complexity measure of set-systems,
or hypergraphs. Its application to graphs is usually done by considering the
sets of neighborhoods of the vertices (cf. Alon et al. (2006) and Chepoi,
Estellon, and Vaxes (2007)), hence providing a set-system. But the graph
structure is lost in the process. The aim of this paper is to introduce the
notion of paired VC-dimension, a generalization of VC-dimension to set-systems
endowed with a graph structure, hence a collection of pairs of subsets.
The classical VC-theory is generally used in combinatorics to bound the
transversality of a hypergraph in terms of its fractional transversality and
its VC-dimension. Similarly, we bound the chromatic number in terms of
fractional transversality and paired VC-dimension. This approach turns out to
be very useful for a class of problems raised by Erd\H{o}s and Simonovits
(1973) asking for H-free graphs with minimum degree at least cn and arbitrarily
high chromatic number, where H is a fixed graph and c a positive constant.
We show how the usual VC-dimension gives a short proof of the fact that
triangle-free graphs with minimum degree at least n/3 have bounded chromatic
number, where is the number of vertices. Using paired VC-dimension, we
prove that if the chromatic number of -free graphs with minimum degree at
least cn is unbounded for some positive c, then it is unbounded for all c<1/3.
In other words, one can find H-free graphs with unbounded chromatic number and
minimum degree arbitrarily close to n/3. These H-free graphs are derived from a
construction of Hajnal. The large chromatic number follows from the Borsuk-Ulam
Theorem
Circular Flows in Planar Graphs
For integers , a \emph{circular -flow} is a flow that takes
values from . The Planar Circular Flow
Conjecture states that every -edge-connected planar graph admits a circular
-flow. The cases and are equivalent to the Four
Color Theorem and Gr\"otzsch's 3-Color Theorem. For , the conjecture
remains open. Here we make progress when and . We prove that (i)
{\em every 10-edge-connected planar graph admits a circular 5/2-flow} and (ii)
{\em every 16-edge-connected planar graph admits a circular 7/3-flow.} The dual
version of statement (i) on circular coloring was previously proved by
Dvo\v{r}\'ak and Postle (Combinatorica 2017), but our proof has the advantages
of being much shorter and avoiding the use of computers for case-checking.
Further, it has new implications for antisymmetric flows. Statement (ii) is
especially interesting because the counterexamples to Jaeger's original
Circular Flow Conjecture are 12-edge-connected nonplanar graphs that admit no
circular 7/3-flow. Thus, the planarity hypothesis of (ii) is essential.Comment: 18 pages, 6 figures, plus 2.5 page appendix with 2 figures, comments
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