35 research outputs found
The Cost of Perfection for Matchings in Graphs
Perfect matchings and maximum weight matchings are two fundamental
combinatorial structures. We consider the ratio between the maximum weight of a
perfect matching and the maximum weight of a general matching. Motivated by the
computer graphics application in triangle meshes, where we seek to convert a
triangulation into a quadrangulation by merging pairs of adjacent triangles, we
focus mainly on bridgeless cubic graphs. First, we characterize graphs that
attain the extreme ratios. Second, we present a lower bound for all bridgeless
cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic
graphs, most of which are shown to be tight. Additionally, we present tight
bounds for the class of regular bipartite graphs
On some intriguing problems in Hamiltonian graph theory -- A survey
We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, -tough graphs, and claw-free graphs
How tough is toughness?
The concept of toughness was introduced by Chvátal [34] more than forty years ago. Toughness resembles vertex connectivity, but is different in the sense that it takes into account what the effect of deleting a vertex cut is on the number of resulting components. As we will see, this difference has major consequences in terms of computational complexity and on the implications with respect to cycle structure, in particular the existence of Hamilton cycles and k-factors
On vertices enforcing a Hamiltonian cycle
A nonempty vertex set X ⊆ V(G) of a hamiltonian graph G is called an of G if every X-cycle of G (i.e. a cycle of G containing all vertices of X) is hamiltonian. The h(G) of a graph G is defined to be the smallest cardinality of an H-force set of G. In the paper the study of this parameter is introduced and its value or a lower bound for outerplanar graphs, planar graphs, k-connected graphs and prisms over graphs is determined
Hamiltonicity of locally hamiltonian and locally traceable graphs
Please read abstract in the article.The University of South Africa and the National Research Foundation of South Africa for their sponsorship of the Salt Rock Workshops of 28 July–10 August 2013 and 20–30 January 2016, which contributed towards results in this paper.
The authors thank the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) for financial support, grant number BA2017/268. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS.
This material is based upon the third author’s work supported by the National Research Foundation of S.A. under Grant number 81075 and the second author’s work supported by the National Research Foundation of S.A. under Grant number 107668.http://www.elsevier.com/locate/dam2019-02-19hj2018Mathematics and Applied Mathematic
A Structured Table of Graphs with Symmetries and Other Special Properties
We organize a table of regular graphs with minimal diameters and minimal mean
path lengths, large bisection widths and high degrees of symmetries, obtained
by enumerations on supercomputers. These optimal graphs, many of which are
newly discovered, may find wide applications, for example, in design of network
topologies.Comment: add details about automorphism grou
On the minimum leaf number of cubic graphs
The \emph{minimum leaf number} of a connected graph is
defined as the minimum number of leaves of the spanning trees of . We
present new results concerning the minimum leaf number of cubic graphs: we show
that if is a connected cubic graph of order , then , improving on the best known result in [Inf. Process.
Lett. 105 (2008) 164-169] and proving the conjecture in [Electron. J. Graph
Theory and Applications 5 (2017) 207-211]. We further prove that if is also
2-connected, then , improving on the best
known bound in [Math. Program., Ser. A 144 (2014) 227-245]. We also present new
conjectures concerning the minimum leaf number of several types of cubic graphs
and examples showing that the bounds of the conjectures are best possible.Comment: 17 page