21,613 research outputs found
EXTREMAL PROBLEMS CONCERNING CYCLES IN GRAPHS AND THEIR COMPLEMENTS
Let Gt(n) be the class of connected graphs on n vertices having the longest cycle of length t and let
G ∈ Gt(n). Woodall (1976) determined the maximum number of edges of G, ε(G) ≤ w(n,t), where
w(n, t) = (n - 1) t/2 - r(t – r - 1)/2 and r = (n - 1 ) - (t - 1) ⎣(n - 1)/(t - 1)⎦. An alternative proof and
characterization of the extremal (edge-maximal) graphs given by Caccetta and Vijayan (1991). The edge-
maximal graphs have the property that their complements are either disconnected or have a cycle going
through each vertex (i.e. they are hamiltonian). This motivates us to investigate connected graphs with
prescribed circumference (length of the longest cycle) having connected complements with cycles . More
specifically, we focus our investigations on :
Let G(n, c, c ) denote the class of connected graphs on n vertices having circumference c and
whose connected complements have circumference c . The problem of interest is that of
determining the bounds of the number of edges of a graph G ∈ G(n, c, c ) and characterize the
extremal graphs of G(n, c, c ).
We discuss the class G(n, c, c ) and present some results for small c. In particular for c = 4 and
c = n - 2, we provide a complete solution.
Key words : extremal graph, circumferenc
An extremal problem on group connectivity of graphs
Let A be an Abelian group, n \u3e 3 be an integer, and ex(n, A) be the maximum integer such that every n-vertex simple graph with at most ex(n, A) edges is not A-connected. In this paper, we study ex(n, A) for IAI \u3e 3 and present lower and upper bounds for 3 \u3c IAI 5. 0 2012 Elsevier Ltd. All rights reserved
Extremal Optimization for Graph Partitioning
Extremal optimization is a new general-purpose method for approximating
solutions to hard optimization problems. We study the method in detail by way
of the NP-hard graph partitioning problem. We discuss the scaling behavior of
extremal optimization, focusing on the convergence of the average run as a
function of runtime and system size. The method has a single free parameter,
which we determine numerically and justify using a simple argument. Our
numerical results demonstrate that on random graphs, extremal optimization
maintains consistent accuracy for increasing system sizes, with an
approximation error decreasing over runtime roughly as a power law t^(-0.4). On
geometrically structured graphs, the scaling of results from the average run
suggests that these are far from optimal, with large fluctuations between
individual trials. But when only the best runs are considered, results
consistent with theoretical arguments are recovered.Comment: 34 pages, RevTex4, 1 table and 20 ps-figures included, related papers
available at http://www.physics.emory.edu/faculty/boettcher
Extremal Optimization at the Phase Transition of the 3-Coloring Problem
We investigate the phase transition of the 3-coloring problem on random
graphs, using the extremal optimization heuristic. 3-coloring is among the
hardest combinatorial optimization problems and is closely related to a 3-state
anti-ferromagnetic Potts model. Like many other such optimization problems, it
has been shown to exhibit a phase transition in its ground state behavior under
variation of a system parameter: the graph's mean vertex degree. This phase
transition is often associated with the instances of highest complexity. We use
extremal optimization to measure the ground state cost and the ``backbone'', an
order parameter related to ground state overlap, averaged over a large number
of instances near the transition for random graphs of size up to 512. For
graphs up to this size, benchmarks show that extremal optimization reaches
ground states and explores a sufficient number of them to give the correct
backbone value after about update steps. Finite size scaling gives
a critical mean degree value . Furthermore, the
exploration of the degenerate ground states indicates that the backbone order
parameter, measuring the constrainedness of the problem, exhibits a first-order
phase transition.Comment: RevTex4, 8 pages, 4 postscript figures, related information available
at http://www.physics.emory.edu/faculty/boettcher
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