12,754 research outputs found
Cycles in Sparse Graphs II
The {\em independence ratio} of a graph is defined by where is the
independence number of the subgraph of induced by . The independence
ratio is a relaxation of the chromatic number in the sense that
for every graph , while for many natural classes of
graphs these quantities are almost equal. In this paper, we address two old
conjectures of Erd\H{o}s on cycles in graphs with large chromatic number and a
conjecture of Erd\H{o}s and Hajnal on graphs with infinite chromatic number.Comment: 16 pages, 1 figur
Generalization of some results on list coloring and DP-coloring
In this work, we introduce DPG-coloring using the concepts of DP-coloring and
variable degeneracy to modify the proofs on the following papers: (i)
DP-3-coloring of planar graphs without , -cycles and cycles of two
lengths from (R. Liu, S. Loeb, M. Rolek, Y. Yin, G. Yu, Graphs
and Combinatorics 35(3) (2019) 695-705), (ii) Every planar graph without
-cycles adjacent simultaneously to -cycles and -cycles is
DP--colorable when (P. Sittitrai, K. Nakprasit,
arXiv:1801.06760(2019) preprint), (iii) Every planar graph is -choosable (C.
Thomassen, J. Combin. Theory Ser. B 62 (1994) 180-181). Using this
modification, we obtain more results on list coloring, DP-coloring,
list-forested coloring, and variable degeneracy.Comment: arXiv admin note: text overlap with arXiv:1807.0081
Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph
We prove that after an arbitrarily small adjustment of edge lengths, the
spectrum of a compact quantum graph with -type vertex conditions can be
simple. We also show that the eigenfunctions, with the exception of those
living entirely on a looping edge, can be made to be non-vanishing on all
vertices of the graph. As an application of the above result, we establish that
the secular manifold (also called "determinant manifold") of a large family of
graphs has exactly two smooth connected components.Comment: 18 pages, 12 figure
Tree-colorable maximal planar graphs
A tree-coloring of a maximal planar graph is a proper vertex -coloring
such that every bichromatic subgraph, induced by this coloring, is a tree. A
maximal planar graph is tree-colorable if has a tree-coloring. In this
article, we prove that a tree-colorable maximal planar graph with
contains at least four odd-vertices.
Moreover, for a tree-colorable maximal planar graph of minimum degree 4 that
contains exactly four odd-vertices, we show that the subgraph induced by its
four odd-vertices is not a claw and contains no triangles.Comment: 18pages,10figure
The domatic number of regular and almost regular graphs
The domatic number of a graph , denoted , is the maximum possible
cardinality of a family of disjoint sets of vertices of , each set being a
dominating set of . It is well known that every graph without isolated
vertices has . For every , it is known that there are graphs
with minimum degree at least and with . In this paper we prove
that this is not the case if is -regular or {\em almost} -regular (by
``almost'' we mean that the minimum degree is and the maximum degree is at
most for some fixed real number ). In this case we prove that
. We also prove that the order of magnitude
cannot be improved. One cannot replace the constant 2 with a constant
smaller than 1. The proof uses the so called {\em semi-random method} which
means that combinatorial objects are generated via repeated applications of the
probabilistic method; in our case iterative applications of the Lov\'asz Local
Lemma.Comment: 10 page
Problem collection from the IML programme: Graphs, Hypergraphs, and Computing
This collection of problems and conjectures is based on a subset of the open
problems from the seminar series and the problem sessions of the Institut
Mitag-Leffler programme Graphs, Hypergraphs, and Computing. Each problem
contributor has provided a write up of their proposed problem and the
collection has been edited by Klas Markstr\"om.Comment: This problem collection is published as part of the IML preprint
series for the research programme and also available there
http://www.mittag-leffler.se/research-programs/preprint-series?course_id=4401.
arXiv admin note: text overlap with arXiv:1403.5975, arXiv:0706.4101 by other
author
Long cycles in Hamiltonian graphs
We prove that if an -vertex graph with minimum degree at least
contains a Hamiltonian cycle, then it contains another cycle of length
; this implies, in particular, that a well-known conjecture of Sheehan
from 1975 holds asymptotically. Our methods, which combine constructive,
poset-based techniques and non-constructive, parity-based arguments, may be of
independent interest.Comment: 15 pages, submitted, some typos fixe
Design of LDPC Codes using Multipath EMD Strategies and Progressive Edge Growth
Low-density parity-check (LDPC) codes are capable of achieving excellent
performance and provide a useful alternative for high performance applications.
However, at medium to high signal-to-noise ratios (SNR), an observable error
floor arises from the loss of independence of messages passed under iterative
graph-based decoding. In this paper, the error floor performance of short block
length codes is improved by use of a novel candidate selection metric in code
graph construction. The proposed Multipath EMD approach avoids harmful
structures in the graph by evaluating certain properties of the cycles which
may be introduced in each edge placement. We present Multipath EMD based
designs for several structured LDPC codes including quasi-cyclic and irregular
repeat accumulate codes. In addition, an extended class of diversity-achieving
codes on the challenging block fading channel is proposed and considered with
the Multipath EMD design. This combined approach is demonstrated to provide
gains in decoder convergence and error rate performance. A simulation study
evaluates the performance of the proposed and existing state-of-the-art
methods.Comment: 18 figures, 28 pages in IEEE Transactions on Communications, 201
Cycle lengths modulo in large 3-connected cubic graphs
We prove that for all natural numbers and where is odd, there
exists a natural number such that any 3-connected cubic graph with at
least vertices contains a cycle of length modulo . We also
construct a family of graphs showing that this is not true for 2-connected
cubic graphs if and are divisible by 3 and
Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs
We study the nodal count of the so-called bi-dendral graphs and show that it
exhibits an anomaly: the nodal surplus is never equal to 0 or , the
first Betti number of the graph. According to the nodal-magnetic theorem, this
means that bands of the magnetic spectrum (dispersion relation) of such graphs
do not have maxima or minima at the "usual" symmetry points of the fundamental
domain of the reciprocal space of magnetic parameters.
In search of the missing extrema we prove a necessary condition for a smooth
critical point to happen inside the reciprocal fundamental domain. Using this
condition, we identify the extrema as the singularities in the dispersion
relation of the maximal abelian cover of the graph (the honeycomb graph being
an important example).
In particular, our results show that the anomalous nodal count is an
indication of the presence of the conical points in the dispersion relation of
the maximal universal cover. Also, we discover that the conical points are
present in the dispersion relation of graphs with much less symmetry than was
required in previous investigations.Comment: 22 pages, 6 figures; corrections suggested by a referee; expanded
interlacing lemma 4.
- …