1,664 research outputs found
On the structure of graphs without short cycles
The objective of this thesis is to study cages, constructions and properties of such families of graphs. For this, the study of graphs without short cycles plays a fundamental role in order to develop some knowledge on their structure, so we can later deal with the problems on cages. Cages were introduced by Tutte in 1947. In 1963, Erdös and Sachs proved that (k, g) -cages exist for any given values of k and g. Since then, large amount of research in cages has been devoted to their construction.
In this work we study structural properties such as the connectivity, diameter, and degree regularity of graphs without short cycles.
In some sense, connectivity is a measure of the reliability of a network. Two graphs with the same edge-connectivity, may be considered to have different reliabilities, as a more refined index than the edge-connectivity, edge-superconnectivity is proposed together with some other parameters called restricted connectivities.
By relaxing the conditions that are imposed for the graphs to be cages, we can achieve more refined connectivity properties on these families and also we have an approach to structural properties of the family of graphs with more restrictions (i.e., the cages).
Our aim, by studying such structural properties of cages is to get a deeper insight into their structure so we can attack the problem of their construction.
By way of example, we studied a condition on the diameter in relation to the girth pair of a graph, and as a corollary we obtained a result guaranteeing restricted connectivity of a special family of graphs arising from geometry, such as polarity graphs.
Also, we obtained a result proving the edge superconnectivity of semiregular cages. Based on these studies it was possible to develop the study of cages.
Therefore obtaining a relevant result with respect to the connectivity of cages, that is, cages are k/2-connected. And also arising from the previous work on girth pairs we obtained constructions for girth pair cages that proves a bound conjectured by Harary and Kovács, relating the order of girth pair cages with the one for cages. Concerning the degree and the diameter, there is the concept of a Moore graph, it was introduced by Hoffman and Singleton after Edward F. Moore, who posed the question of describing and classifying these graphs.
As well as having the maximum possible number of vertices for a given combination of degree and diameter, Moore graphs have the minimum possible number of vertices for a regular graph with given degree and girth. That is, any Moore graph is a cage. The formula for the number of vertices in a Moore graph can be generalized to allow a definition of Moore graphs with even girth (bipartite Moore graphs) as well as odd girth, and again these graphs are cages. Thus, Moore graphs give a lower bound for the order of cages, but they are known to exist only for very specific values of k, therefore it is interesting to study how far a cage is from this bound, this value is called the excess of a cage.
We studied the excess of graphs and give a contribution, in the sense of the work of Biggs and Ito, relating the bipartition of girth 6 cages with their orders. Entire families of cages can be obtained from finite geometries, for example, the graphs of incidence of projective planes of order q a prime power, are (q+1, 6)-cages. Also by using other incidence structures such as the generalized quadrangles or generalized hexagons, it can be obtained families of cages of girths 8 and 12.
In this thesis, we present a construction of an entire family of girth 7 cages that arises from some combinatorial properties of the incidence graphs of generalized quadrangles of order (q,q)
On Rainbow Connection Number and Connectivity
Rainbow connection number, , of a connected graph is the minimum
number of colours needed to colour its edges, so that every pair of vertices is
connected by at least one path in which no two edges are coloured the same. In
this paper we investigate the relationship of rainbow connection number with
vertex and edge connectivity. It is already known that for a connected graph
with minimum degree , the rainbow connection number is upper bounded by
[Chandran et al., 2010]. This directly gives an upper
bound of and for rainbow
connection number where and , respectively, denote the edge
and vertex connectivity of the graph. We show that the above bound in terms of
edge connectivity is tight up-to additive constants and show that the bound in
terms of vertex connectivity can be improved to , for any . We conjecture that rainbow connection
number is upper bounded by and show that it is true for
. We also show that the conjecture is true for chordal graphs and
graphs of girth at least 7.Comment: 10 page
Cubic graphs with large circumference deficit
The circumference of a graph is the length of a longest cycle. By
exploiting our recent results on resistance of snarks, we construct infinite
classes of cyclically -, - and -edge-connected cubic graphs with
circumference ratio bounded from above by , and
, respectively. In contrast, the dominating cycle conjecture implies
that the circumference ratio of a cyclically -edge-connected cubic graph is
at least .
In addition, we construct snarks with large girth and large circumference
deficit, solving Problem 1 proposed in [J. H\"agglund and K. Markstr\"om, On
stable cycles and cycle double covers of graphs with large circumference, Disc.
Math. 312 (2012), 2540--2544]
On Linkedness of Cartesian Product of Graphs
We study linkedness of Cartesian product of graphs and prove that the product
of an -linked and a -linked graphs is -linked if the graphs are
sufficiently large. Further bounds in terms of connectivity are shown. We
determine linkedness of product of paths and product of cycles
Hierarchical and High-Girth QC LDPC Codes
We present a general approach to designing capacity-approaching high-girth
low-density parity-check (LDPC) codes that are friendly to hardware
implementation. Our methodology starts by defining a new class of
"hierarchical" quasi-cyclic (HQC) LDPC codes that generalizes the structure of
quasi-cyclic (QC) LDPC codes. Whereas the parity check matrices of QC LDPC
codes are composed of circulant sub-matrices, those of HQC LDPC codes are
composed of a hierarchy of circulant sub-matrices that are in turn constructed
from circulant sub-matrices, and so on, through some number of levels. We show
how to map any class of codes defined using a protograph into a family of HQC
LDPC codes. Next, we present a girth-maximizing algorithm that optimizes the
degrees of freedom within the family of codes to yield a high-girth HQC LDPC
code. Finally, we discuss how certain characteristics of a code protograph will
lead to inevitable short cycles, and show that these short cycles can be
eliminated using a "squashing" procedure that results in a high-girth QC LDPC
code, although not a hierarchical one. We illustrate our approach with designed
examples of girth-10 QC LDPC codes obtained from protographs of one-sided
spatially-coupled codes.Comment: Submitted to IEEE Transactions on Information THeor
Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44
The family of snarks -- connected bridgeless cubic graphs that cannot be
3-edge-coloured -- is well-known as a potential source of counterexamples to
several important and long-standing conjectures in graph theory. These include
the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's
conjecture, and several others. One way of approaching these conjectures is
through the study of structural properties of snarks and construction of small
examples with given properties. In this paper we deal with the problem of
determining the smallest order of a nontrivial snark (that is, one which is
cyclically 4-edge-connected and has girth at least 5) of oddness at least 4.
Using a combination of structural analysis with extensive computations we prove
that the smallest order of a snark with oddness at least 4 and cyclic
connectivity 4 is 44. Formerly it was known that such a snark must have at
least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468--488] and one such
snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin.
22 (2015), #P1.51]. The proof requires determining all cyclically
4-edge-connected snarks on 36 vertices, which extends the previously compiled
list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc.
cit.]. As a by-product, we use this new list to test the validity of several
conjectures where snarks can be smallest counterexamples.Comment: 21 page
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