Tracking Data on Mobile Devices

Tato práce pojednává o možnostech sledování pohybu dat na mobilních zařízeních. Rozšiřuje čtenářovo povědomí o rozdělení trhu mobilních operačních systémů a důležitosti bezpečnostních řešení v podnikové sféře. Díky porovnání možností vývoje a bezpečnostních prvků platforem je ucelena představa o důležitosti těchto systémů. Součástí práce jsou vyvinuté aplikace na platformách Android a BlackBerry OS pro záznam sledovaných dat, serverová aplikace pro jejich sběr a iOS aplikace na PhoneLogs pro jejich hromadné zobrazení. Přístup k záznamům je možný až po úspěšné autentizaci, jednotlivé databáze jsou šifrovány a komunikace je zabezpečená SSL certifikátem.This thesis researches the possibilities of tracking mobile device data and their movement. It encompasses the division of the mobile operating systems market and also the importance of security solutions in business sector. Thanks to comparing the possibilities of development and security elements of individual platforms, the overview of importance is completed. The thesis also contains developed applications on Android and BlackBerry OS platforms for recording tracked data, a server application for their collection and a iOS application on PhoneLogs for their wholesale viewing. Accessing these records is possible after a successful authentication, individual databases are encrypted and communication secured with an SSL certificate.

Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44

The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one which is cyclically 4-edge-connected and has girth at least 5) of oddness at least 4. Using a combination of structural analysis with extensive computations we prove that the smallest order of a snark with oddness at least 4 and cyclic connectivity 4 is 44. Formerly it was known that such a snark must have at least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468--488] and one such snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin. 22 (2015), #P1.51]. The proof requires determining all cyclically 4-edge-connected snarks on 36 vertices, which extends the previously compiled list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc. cit.]. As a by-product, we use this new list to test the validity of several conjectures where snarks can be smallest counterexamples.Comment: 21 page

Regular Embeddings of Canonical Double Coverings of Graphs

AbstractThis paper addresses the question of determining, for a given graphG, all regular maps havingGas their underlying graph, i.e., all embeddings ofGin closed surfaces exhibiting the highest possible symmetry. We show that ifGsatisfies certain natural conditions, then all orientable regular embeddings of its canonical double covering, isomorphic to the tensor productG⊗K2, can be described in terms of regular embeddings ofG. This allows us to “lift” the classification of regular embeddings of a given graph to a similar classification for its canonical double covering and to establish various properties of the “derived” maps by employing those of the “base” maps. We apply these results to determining all orientable regular embeddings of the tensor productsKn⊗K2(known as the cocktail-party graphs) and of then-dipolesDn, the graphs consisting of two vertices and n parallel edges joining them. In the first case we show, in particular, that regular embeddings ofKn⊗K2exist only ifnis a prime powerpl, and there are 2φ(n−1) orφ(n−1) isomorphism classes of such maps (whereφis Euler's function) according to whetherlis even or odd. Forleven an interesting new infinite family of regular maps is discovered. In the second case, orientable regular embeddings ofDnexist for each positive integern, and their number is a power of 2 depending on the decomposition ofninto primes

Regular Maps on Surfaces with Large Planar Width

AbstractA map is a cell decomposition of a closed surface; it is regular if its automorphism group acts transitively on the flags, mutually incident vertex-edge-face triples. The main purpose of this paper is to establish, by elementary methods, the following result: for each positive integer w and for each pair of integersp≥ 3 and q≥ 3 satisfying 1/p+ 1/q≤ 1/2, there is an orientable regular map with face-size p and valency q such that every non-contractible simple closed curve on the surface meets the 1-skeleton of the map in at least w points. This result has several interesting consequences concerning maps on surfaces, graphs and related concepts. For example, MacBeath’s theorem about the existence of infinitely many Hurwitz groups, or Vince’s theorem about regular maps of given type (p, q), or residual finiteness of triangle groups, all follow from our result