2,028 research outputs found
Hamilton cycles in dense vertex-transitive graphs
A famous conjecture of Lov\'asz states that every connected vertex-transitive
graph contains a Hamilton path. In this article we confirm the conjecture in
the case that the graph is dense and sufficiently large. In fact, we show that
such graphs contain a Hamilton cycle and moreover we provide a polynomial time
algorithm for finding such a cycle.Comment: 26 pages, 3 figures; referees' comments incorporated; accepted for
publication in Journal of Combinatorial Theory, series
Distributed CONGEST Algorithm for Finding Hamiltonian Paths in Dirac Graphs and Generalizations
We study the problem of finding a Hamiltonian cycle under the promise that
the input graph has a minimum degree of at least , where denotes the
number of vertices in the graph. The classical theorem of Dirac states that
such graphs (a.k.a. Dirac graphs) are Hamiltonian, i.e., contain a Hamiltonian
cycle. Moreover, finding a Hamiltonian cycle in Dirac graphs can be done in
polynomial time in the classical centralized model.
This paper presents a randomized distributed CONGEST algorithm that finds
w.h.p. a Hamiltonian cycle (as well as maximum matching) within
rounds under the promise that the input graph is a Dirac graph. This upper
bound is in contrast to general graphs in which both the decision and search
variants of Hamiltonicity require rounds, as shown by
Bachrach et al. [PODC'19].
In addition, we consider two generalizations of Dirac graphs: Ore graphs and
Rahman-Kaykobad graphs [IPL'05]. In Ore graphs, the sum of the degrees of every
pair of non-adjacent vertices is at least , and in Rahman-Kaykobad graphs,
the sum of the degrees of every pair of non-adjacent vertices plus their
distance is at least . We show how our algorithm for Dirac graphs can be
adapted to work for these more general families of graphs
JGraphT -- A Java library for graph data structures and algorithms
Mathematical software and graph-theoretical algorithmic packages to
efficiently model, analyze and query graphs are crucial in an era where
large-scale spatial, societal and economic network data are abundantly
available. One such package is JGraphT, a programming library which contains
very efficient and generic graph data-structures along with a large collection
of state-of-the-art algorithms. The library is written in Java with stability,
interoperability and performance in mind. A distinctive feature of this library
is the ability to model vertices and edges as arbitrary objects, thereby
permitting natural representations of many common networks including
transportation, social and biological networks. Besides classic graph
algorithms such as shortest-paths and spanning-tree algorithms, the library
contains numerous advanced algorithms: graph and subgraph isomorphism; matching
and flow problems; approximation algorithms for NP-hard problems such as
independent set and TSP; and several more exotic algorithms such as Berge graph
detection. Due to its versatility and generic design, JGraphT is currently used
in large-scale commercial, non-commercial and academic research projects. In
this work we describe in detail the design and underlying structure of the
library, and discuss its most important features and algorithms. A
computational study is conducted to evaluate the performance of JGraphT versus
a number of similar libraries. Experiments on a large number of graphs over a
variety of popular algorithms show that JGraphT is highly competitive with
other established libraries such as NetworkX or the BGL.Comment: Major Revisio
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