1,084 research outputs found
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
Construction of cycle double covers for certain classes of graphs
We introduce two classes of graphs, Indonesian graphs and -doughnut graphs. Cycle double covers are constructed for these classes. In case of doughnut graphs this is done for the values and 4
On duality and fractionality of multicommodity flows in directed networks
In this paper we address a topological approach to multiflow (multicommodity
flow) problems in directed networks. Given a terminal weight , we define a
metrized polyhedral complex, called the directed tight span , and
prove that the dual of -weighted maximum multiflow problem reduces to a
facility location problem on . Also, in case where the network is
Eulerian, it further reduces to a facility location problem on the tropical
polytope spanned by . By utilizing this duality, we establish the
classifications of terminal weights admitting combinatorial min-max relation
(i) for every network and (ii) for every Eulerian network. Our result includes
Lomonosov-Frank theorem for directed free multiflows and
Ibaraki-Karzanov-Nagamochi's directed multiflow locking theorem as special
cases.Comment: 27 pages. Fixed minor mistakes and typos. To appear in Discrete
Optimizatio
Infinity Algebras and the Homology of Graph Complexes
An A-infinity algebra is a generalization of a associative algebra, and an
L-infinity algebra is a generalization of a Lie algebra. In this paper, we show
that an L-infinity algebra with an invariant inner product determines a cycle
in the homology of the complex of metric ordinary graphs. Since the cyclic
cohomology of a Lie algebra with an invariant inner product determines
infinitesimal deformations of the Lie algebra into an L-infinity algebra with
an invariant inner product, this construction shows that a cyclic cocycle of a
Lie algebra determines a cycle in the homology of the graph complex. In this
paper a simple proof of the corresponding result for A-infinity algebras, which
was proved in a different manner in an earlier paper, is given.Comment: 14 pages, amslatex document, 4 figure
Functional limit theorems for random regular graphs
Consider d uniformly random permutation matrices on n labels. Consider the
sum of these matrices along with their transposes. The total can be interpreted
as the adjacency matrix of a random regular graph of degree 2d on n vertices.
We consider limit theorems for various combinatorial and analytical properties
of this graph (or the matrix) as n grows to infinity, either when d is kept
fixed or grows slowly with n. In a suitable weak convergence framework, we
prove that the (finite but growing in length) sequences of the number of short
cycles and of cyclically non-backtracking walks converge to distributional
limits. We estimate the total variation distance from the limit using Stein's
method. As an application of these results we derive limits of linear
functionals of the eigenvalues of the adjacency matrix. A key step in this
latter derivation is an extension of the Kahn-Szemer\'edi argument for
estimating the second largest eigenvalue for all values of d and n.Comment: Added Remark 27. 39 pages. To appear in Probability Theory and
Related Field
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