252 research outputs found
Embedding Digraphs on Orientable Surfaces
AbstractWe consider a notion of embedding digraphs on orientable surfaces, applicable to digraphs in which the indegree equals the outdegree for every vertex, i.e., Eulerian digraphs. This idea has been considered before in the context of compatible Euler tours or orthogonal A-trails by Andersen and by Bouchet. This prior work has mostly been limited to embeddings of Eulerian digraphs on predetermined surfaces and to digraphs with underlying graphs of maximum degree at most 4. In this paper, a foundation is laid for the study of all Eulerian digraph embeddings. Results are proved which are analogous to those fundamental to the theory of undirected graph embeddings, such as Duke's theorem [5], and an infinite family of digraphs which demonstrates that the genus range for an embeddable digraph can be any nonnegative integer given. We show that it is possible to have genus range equal to one, with arbitrarily large minimum genus, unlike in the undirected case. The difference between the minimum genera of a digraph and its underlying graph is considered, as is the difference between the maximum genera. We say that a digraph is upper-embeddable if it can be embedded with two or three regions and prove that every regular tournament is upper-embeddable
On the linear algebra of local complementation
AbstractWe explore the connections between the linear algebra of symmetric matrices over GF(2) and the circuit theory of 4-regular graphs. In particular, we show that the equivalence relation on simple graphs generated by local complementation can also be generated by an operation defined using inverse matrices
Generation and Properties of Snarks
For many of the unsolved problems concerning cycles and matchings in graphs
it is known that it is sufficient to prove them for \emph{snarks}, the class of
nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part
of this paper we present a new algorithm for generating all non-isomorphic
snarks of a given order. Our implementation of the new algorithm is 14 times
faster than previous programs for generating snarks, and 29 times faster for
generating weak snarks. Using this program we have generated all non-isomorphic
snarks on vertices. Previously lists up to vertices have been
published. In the second part of the paper we analyze the sets of generated
snarks with respect to a number of properties and conjectures. We find that
some of the strongest versions of the cycle double cover conjecture hold for
all snarks of these orders, as does Jaeger's Petersen colouring conjecture,
which in turn implies that Fulkerson's conjecture has no small counterexamples.
In contrast to these positive results we also find counterexamples to eight
previously published conjectures concerning cycle coverings and the general
cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated
and typos corrected. This version differs from the published one in that the
Arxiv-version has data about the automorphisms of snarks; Journal of
Combinatorial Theory. Series B. 201
Algorithmic Problems Arising in Posets and Permutations
Partially ordered sets and permutations are combinatorial structures having vast applications in theoretical computer science. In this thesis, we study various computational and algorithmic problems related to these structures. The first chapter of the thesis contains discussion about randomized fully polynomial approximation schemes obtained by employing Markov chain Monte Carlo. In this chapter we study various Markov chains that we call: the gladiator chain, the interval chain, and cube shuffling. Our objective is to identify some conditions that assure rapid mixing; and we obtain partial results. The gladiator chain is a biased random walk on the set of permutations. This chain is related to self organizing lists, and various versions of it have been studied. The interval chain is a random walk on the set of points in whose coordinates respect a partial order. Since the sample space of the interval chain is continuous, many mixing techniques for discrete chains are not applicable to it. The cube shuffle chain is a generalization of H\r{a}stad\u27s square shuffle. The importance of this chain is that it mixes in constant number of steps. In the second chapter, we are interested in calculating expected value of real valued function on a set of combinatorial structures , given a probability distribution on it. We first suggest a Markov chain Monte Carlo approach to this problem. We identify the conditions under which our proposed solution will be efficient, and present examples where it fails. Then, we study homomesy. Homomesy is a phenomenon introduced by Jim Propp and Tom Roby. We say the triple ( is a permutation mapping to itself) exhibits homomesy, if the average of along all -orbits of is a constant only depending on and . We study homomesy and obtain some results when is the set of ideals in a class of simply described lattices
A hálózatelemzés elmélete és rendőrségi gyakorlata
The aim of our study is to highlight the usefulness of network analysis methods in the field of law enforcement, and that the future of this field depends on collaboration and productive cooperation of theory and practice. After a brief historical overview, we introduce the most important concepts related to network analysis and graph theory, and then present a possible resolution strategy for a mafia network. Finally, we provide an overview of the directions of network analysis that can be used in profiling.Tanulmányunk cĂ©lja, hogy rávilágĂtson a hálĂłzatelemzĂ©si mĂłdszerek hasznosságára a rendvĂ©delem terĂĽletĂ©n, illetve arra, hogy e terĂĽlet jövĹ‘je az elmĂ©let Ă©s a gyakorlat összefogásán, Ă©s produktĂv egyĂĽttműködĂ©sĂ©n mĂşlik. Rövid törtĂ©neti áttekintĂ©s után ismertetjĂĽk a hálĂłzatelemzĂ©ssel Ă©s gráfelmĂ©lettel kapcsolatos legfontosabb fogalmakat, majd bemutatjuk egy maffiahálĂłzat lehetsĂ©ges bomlasztási stratĂ©giáját. VĂ©gezetĂĽl elmĂ©leti szinten taglaljuk a hálĂłzatelemzĂ©s Ă©s a profilalkotás kapcsolĂłdási pontjait
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