5,478 research outputs found
Random cubic graphs are not homomorphic to the cycle of size 7
We prove that a random cubic graph almost surely is not homomorphic to a
cycle of size 7. This implies that there exist cubic graphs of arbitrarily high
girth with no homomorphisms to the cycle of size 7
On Colorings of Graph Powers
In this paper, some results concerning the colorings of graph powers are
presented. The notion of helical graphs is introduced. We show that such graphs
are hom-universal with respect to high odd-girth graphs whose st power
is bounded by a Kneser graph. Also, we consider the problem of existence of
homomorphism to odd cycles. We prove that such homomorphism to a -cycle
exists if and only if the chromatic number of the st power of
is less than or equal to 3, where is the 2-subdivision of . We also
consider Ne\v{s}et\v{r}il's Pentagon problem. This problem is about the
existence of high girth cubic graphs which are not homomorphic to the cycle of
size five. Several problems which are closely related to Ne\v{s}et\v{r}il's
problem are introduced and their relations are presented
Distributed Distance- Dominating Set on Sparse High-Girth Graphs
The dominating set problem and its generalization, the distance- dominating set problem, are among the well-studied problems in the sequential settings. In distributed models of computation, unlike for domination, not much is known about distance-r domination. This is actually the case for other important closely-related covering problem, namely, the distance- independent set problem. By result of Kuhn et al. we know the distributed domination problem is hard on high girth graphs; we study the problem on a slightly restricted subclass of these graphs: graphs of bounded expansion with high girth, i.e. their girth should be at least . We show that in such graphs, for every constant , a simple greedy CONGEST algorithm provides a constant-factor approximation of the minimum distance- dominating set problem, in a constant number of rounds. More precisely, our constants are dependent to , not to the size of the graph. This is the first algorithm that shows there are non-trivial constant factor approximations in constant number of rounds for any distance -covering problem in distributed settings. To show the dependency on r is inevitable, we provide an unconditional lower bound showing the same problem is hard already on rings. We also show that our analysis of the algorithm is relatively tight, that is any significant improvement to the approximation factor requires new algorithmic ideas
High-Girth Near-Ramanujan Graphs with Lossy Vertex Expansion
Kahale proved that linear sized sets in -regular Ramanujan graphs have
vertex expansion and complemented this with construction of
near-Ramanujan graphs with vertex expansion no better than .
However, the construction of Kahale encounters highly local obstructions to
better vertex expansion. In particular, the poorly expanding sets are
associated with short cycles in the graph. Thus, it is natural to ask whether
high-girth Ramanujan graphs have improved vertex expansion. Our results are
two-fold:
1. For every for prime and infinitely many , we exhibit an
-vertex -regular graph with girth and vertex
expansion of sublinear sized sets bounded by whose nontrivial
eigenvalues are bounded in magnitude by .
2. In any Ramanujan graph with girth , all sets of size bounded by
have vertex expansion .
The tools in analyzing our construction include the nonbacktracking operator
of an infinite graph, the Ihara--Bass formula, a trace moment method inspired
by Bordenave's proof of Friedman's theorem, and a method of Kahale to study
dispersion of eigenvalues of perturbed graphs.Comment: 15 pages, 1 figur
Bridge Girth: A Unifying Notion in Network Design
A classic 1993 paper by Alth\H{o}fer et al. proved a tight reduction from
spanners, emulators, and distance oracles to the extremal function of
high-girth graphs. This paper initiated a large body of work in network design,
in which problems are attacked by reduction to or the analogous
extremal function for other girth concepts. In this paper, we introduce and
study a new girth concept that we call the bridge girth of path systems, and we
show that it can be used to significantly expand and improve this web of
connections between girth problems and network design. We prove two kinds of
results:
1) We write the maximum possible size of an -node, -path system with
bridge girth as , and we write a certain variant for
"ordered" path systems as . We identify several arguments in
the literature that implicitly show upper or lower bounds on ,
and we provide some polynomially improvements to these bounds. In particular,
we construct a tight lower bound for , and we polynomially
improve the upper bounds for and .
2) We show that many state-of-the-art results in network design can be
recovered or improved via black-box reductions to or .
Examples include bounds for distance/reachability preservers, exact hopsets,
shortcut sets, the flow-cut gaps for directed multicut and sparsest cut, an
integrality gap for directed Steiner forest.
We believe that the concept of bridge girth can lead to a stronger and more
organized map of the research area. Towards this, we leave many open problems,
related to both bridge girth reductions and extremal bounds on the size of path
systems with high bridge girth
Analysis Of The Girth For Regular Bi-partite Graphs With Degree 3
The goal of this paper is to derive the detailed description of the
Enumeration Based Search Algorithm from the high level description provided in
[16], analyze the experimental results from our implementation of the
Enumeration Based Search Algorithm for finding a regular bi-partite graph of
degree 3, and compare it with known results from the available literature. We
show that the values of m for a given girth g for (m, 3) BTUs are within the
known mathematical bounds for regular bi-partitite graphs from the available
literature
A Breezing Proof of the KMW Bound
In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW)
proved a hardness result for several fundamental graph problems in the LOCAL
model: For any (randomized) algorithm, there are input graphs with nodes
and maximum degree on which (expected) communication rounds are
required to obtain polylogarithmic approximations to a minimum vertex cover,
minimum dominating set, or maximum matching. Via reduction, this hardness
extends to symmetry breaking tasks like finding maximal independent sets or
maximal matchings. Today, more than years later, there is still no proof
of this result that is easy on the reader. Setting out to change this, in this
work, we provide a fully self-contained and proof of the KMW
lower bound. The key argument is algorithmic, and it relies on an invariant
that can be readily verified from the generation rules of the lower bound
graphs.Comment: 21 pages, 6 figure
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