We prove the undecidability of Core XPath 1.0 (CXP) [G. Gottlob and C. Koch,inProc. of 17th Ann. IEEE Symp. on Logic in Computer Science,LICS ’02(Copenhagen,July 2002). IEEE CS Press (2002) 189–202.] extended with anInflationary Fixed Point (IFP) operator. More specifically,we provethat the satisfiability problem of this language is undecidable. In fact,the fragment ofCXP+IFP containing only the self and descendant axes is already undecidable.

In this paper, we prove several new Turán density results for 3-graphs with independent neighbourhoods. We show:

\begin{align*}\pi (K_4^-, C_5, F_{3,2})=12/49, \pi (K_4^-, F_{3,2})=5/18 \textrm {and} \pi (J_4, F_{3,2})=\pi (J_5, F_{3,2})=3/8,\end{align*}

$\binom{|A|}{2}$

\begin{align*}\pi (F_{3,2}, \textrm { induced }K_4^-)=3/8 \textrm {and} \pi (K_5, 5\textrm {-set spanning exactly 8 edges})=3/4.\end{align*}

\begin{align*}\pi (K_4, 4\textrm {-set spanning exactly 1 edge})=5/9.\end{align*}

E. Thorp introduced the following card shuffling model. Suppose the number of cards is even. Cut the deck into two equal piles, then interleave them as follows. Choose the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then choose from the other pile. Continue this way, flipping an independent coin for each pair, until both piles are empty.

We prove an upper bound of O(d3) for the mixing time of the Thorp shuffle with 2d cards, improving on the best known bound of O(d4). As a consequence, we obtain an improved bound on the time required to encrypt a binary message of length d using the Thorp shuffle.

A perfect Kt-matching in a graph G is a spanning subgraph consisting of vertex-disjoint copies of Kt. A classic theorem of Hajnal and Szemerédi states that if G is a graph of order n with minimum degree δ(G) ≥ (t − 1)n/t and t|n, then G contains a perfect Kt-matching. Let G be a t-partite graph with vertex classes V1, …, Vt each of size n. We show that, for any γ > 0, if every vertex x ∈ Vi is joined to at least $\bigl ((t-1)/t + \gamma \bigr )n$ vertices of Vj for each j ≠ i, then G contains a perfect Kt-matching, provided n is large enough. Thus, we verify a conjecture of Fischer [6] asymptotically. Furthermore, we consider a generalization to hypergraphs in terms of the codegree.

A Message Authentication Code (MAC) is a function that takes a message and a key asparameters and outputs an authentication of the message. MAC are used to guarantee thelegitimacy of messages exchanged through a network, since generating a correctauthentication requires the knowledge of the key defined secretly by trusted parties.However, an attacker with access to a sufficiently large number of message/authenticationpairs may use a brute force algorithm to infer the secret key: from a set containinginitially all possible key candidates, subsequently remove those that yield an incorrectauthentication, proceeding this way for each intercepted message/authentication pair untila single key remains. In this paper, we determine an exact formula for the expected numberof message/authentication pairs that must be used before such form of attack issuccessful, along with an asymptotical bound that is both simple and tight. We conclude byillustrating a modern application where this bound comes in handy, namely the estimationof security levels in reflection-based verification of software integrity.

We introduce a type of isomorphism among strategic games that we call localisomorphism. Local isomorphisms is a weaker version of the notions of strongand weak game isomorphism introduced in [J. Gabarro, A. Garcia and M. Serna,Theor. Comput. Sci. 412 (2011) 6675–6695]. In a localisomorphism it is required to preserve, for any player, the player’s preferences on thesets of strategy profiles that differ only in the action selected by this player. We showthat the game isomorphism problem for local isomorphism is equivalent to the same problemfor strong or weak isomorphism for strategic games given in: general, extensive andformula general form. As a consequence of the results in [J. Gabarro, A. Garcia and M.Serna, Theor. Comput. Sci. 412 (2011) 6675–6695] thisimplies that local isomorphism problem for strategic games is equivalent to (a) thecircuit isomorphism problem for games given in general form, (b) the boolean formulaisomorphism problem for formula games in general form, and (c) the graph isomorphismproblem for games given in explicit form.

We investigate the Sandpile Model and Chip Firing Game and an extension of these modelson cycle graphs. The extended model consists of allowing a negative number of chips ateach vertex. We give the characterization of reachable configurations and of fixed pointsof each model. At the end, we give explicit formula for the number of their fixedpoints.

We consider the problem of sums of dilates in groups of prime order. It is well known that sets with small density and small sumset in p behave like integer sets. Thus, given A ⊂ p of sufficiently small density, it is straightforward to show that

\begin{linenomath}$$| \lambda_{1}A+\lambda_{2}A+\cdots+ \lambda_{k}A | \ge\biggl(\sum_{i}|\lambda_{i}|\biggr)|A|- o(|A|).$$\end{linenomath}

A 3-graph is said to contain a generalized 4-cycle if it contains 4 edges A, B, C, D such that A ∩ B=C ∩ D =∅ and A ∪ B=C ∪ D. We show that a 3-graph in which every pair of vertices is contained in at least 4 edges must contain a generalized 4-cycle. When the number of vertices, n, is equivalent to 1 or 5 modulo 20, this result is optimum, in the sense that for such n there are 3-graphs where every pair of vertices is contained in 3 edges but which do not contain a generalized 4-cycle.

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices is j-independent if its intersection with any edge has size strictly less than j. The Kruskal–Katona theorem implies that in an r-uniform hypergraph with a fixed size and order, the hypergraph with the most r-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number of j-independent sets in an r-uniform hypergraph.

It is well known that an intersecting family of subsets of an n-element set can contain at most 2n−1 sets. It is natural to wonder how ‘close’ to intersecting a family of size greater than 2n−1 can be. Katona, Katona and Katona introduced the idea of a ‘most probably intersecting family’. Suppose that is a family and that 0 < p < 1. Let (p) be the (random) family formed by selecting each set in independently with probability p. A family is most probably intersecting if it maximizes the probability that (p) is intersecting over all families of size ||.

Katona, Katona and Katona conjectured that there is a nested sequence consisting of most probably intersecting families of every possible size. We show that this conjecture is false for every value of p provided that n is sufficiently large.

The discrete cube {0, 1}d is a fundamental combinatorial structure. A subcube of {0, 1}d is a subset of 2k of its points formed by fixing k coordinates and allowing the remaining d − k to vary freely. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no r + 1 of them have non-empty intersection, how many pairwise intersections can we have? How many subcubes can we have if among them there are no k which have non-empty intersection and no l which are pairwise disjoint?

These questions are naturally expressed using intersection graphs. The intersection graph of a family of sets has one vertex for each set in the family with two vertices being adjacent if the corresponding subsets intersect. Let $\I(n,d)$ be the set of all n vertex graphs which can be represented as the intersection graphs of subcubes in {0, 1}d. With this notation our first question above asks for the largest number of edges in a Kr+1-free graph in $\I(n,d)$. As such it is a Turán-type problem. We answer this question asymptotically for some ranges of r and d. More precisely we show that if $(k+1)2^{\lfloor\frac{d}{k+1}\rfloor}<n\leq k2^{\lfloor\frac{d}{k}\rfloor}$ for some integer k ≥ 2 then the maximum edge density is $\bigl(1-\frac{1}{k}-o(1)\bigr)$ provided that n is not too close to the lower limit of the range.

The second question can be thought of as a Ramsey-type problem. The maximum such n can be defined in the same way as the usual Ramsey number but only considering graphs which are in $\I(n,d)$. We give bounds for this maximum n mainly concentrating on the case that l is fixed, and make some comparisons with the usual Ramsey number.

The study of the phase transition of random graph processes, and recently in particular Achlioptas processes, has attracted much attention. Achlioptas, D'Souza and Spencer (Science, 2009) gave strong numerical evidence that a variety of edge-selection rules in Achlioptas processes exhibit a discontinuous phase transition. However, Riordan and Warnke (Science, 2011) recently showed that all these processes have a continuous phase transition.

In this work we prove discontinuous phase transitions for three random graph processes: all three start with the empty graph on n vertices and, depending on the process, we connect in every step (i) one vertex chosen randomly from all vertices and one chosen randomly from a restricted set of vertices, (ii) two components chosen randomly from the set of all components, or (iii) a randomly chosen vertex and a randomly chosen component.

Böttcher, Schacht and Taraz (Math. Ann., 2009) gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobás and Komlós (Combin. Probab. Comput., 1999). We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph G on n vertices that forces G to contain every bipartite graph H on n vertices of bounded degree and of bandwidth o(n). This also implies an Ore-type result. In fact, we prove a much stronger result where the condition on G is relaxed to a certain robust expansion property. Our result also confirms the bipartite case of a conjecture of Balogh, Kostochka and Treglown concerning the degree sequence of a graph which forces a perfect H-packing.

In automata theory, quantum computation has been widely examined for finite statemachines, known as quantum finite automata (QFAs), and less attention has been given toQFAs augmented with counters or stacks. In this paper, we focus on such generalizations ofQFAs where the input head operates in one-way or realtime mode, and present some newresults regarding their superiority over their classical counterparts. Our first result isabout the nondeterministic acceptance mode: Each quantum model architecturallyintermediate between realtime finite state automaton and one-way pushdown automaton(one-way finite automaton, realtime and one-way finite automata with one-counter, andrealtime pushdown automaton) is superior to its classical counterpart. The second andthird results are about bounded error language recognition: for anyk > 0, QFAs with k blind counters outperform theirdeterministic counterparts; and, a one-way QFA with a single head recognizes an infinitefamily of languages, which can be recognized by one-way probabilistic finite automata withat least two heads. Lastly, we compare the nondeterminictic and deterministic acceptancemodes for classical finite automata with k blind counter(s), and we showthat for any k > 0, the nondeterministic models outperform thedeterministic ones.

In this paper, we define k-counting automata as recognizers forω-languages, i.e. languages of infinite words. Weprove that the class of ω-languages they recognize is a proper extensionof the ω-regular languages. In addition we prove that languagesrecognized by k-counting automata are closed under Boolean operations. Itremains an open problem whether or not emptiness is decidable fork-counting automata. However, we conjecture strongly that it is decidableand give formal reasons why we believe so.