148 research outputs found

    Pegging Graphs Yields a Small Diameter

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    We consider the following process for generating large random cubic graphs. Starting with a given graph, repeatedly add edges that join the midpoints of two randomly chosen edges. We show that the growing graph asymptotically almost surely has logarithmic diameter. This process is motivated by a particular type of peer-to-peer network. Our method extends to similar processes that generate regular graphs of higher degre

    Cup Stacking in Graphs

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    Here we introduce a new game on graphs, called cup stacking, following a line of what can be considered as 00-, 11-, or 22-person games such as chip firing, percolation, graph burning, zero forcing, cops and robbers, graph pebbling, and graph pegging, among others. It can be more general, but the most basic scenario begins with a single cup on each vertex of a graph. For a vertex with kk cups on it we can move all its cups to a vertex at distance kk from it, provided the second vertex already has at least one cup on it. The object is to stack all cups onto some pre-described target vertex. We say that a graph is stackable if this can be accomplished for all possible target vertices. In this paper we study cup stacking on many families of graphs, developing a characterization of stackability in graphs and using it to prove the stackability of complete graphs, paths, cycles, grids, the Petersen graph, many Kneser graphs, some trees, cubes of dimension up to 20, "somewhat balanced" complete tt-partite graphs, and Hamiltonian diameter two graphs. Additionally we use the Gallai-Edmonds Structure Theorem, the Edmonds Blossom Algorithm, and the Hungarian algorithm to devise a polynomial algorithm to decide if a diameter two graph is stackable. Our proof that cubes up to dimension 20 are stackable uses Kleitman's Symmetric Chain Decomposition and the new result of Merino, M\"utze, and Namrata that all generalized Johnson graphs (excluding the Petersen graph) are Hamiltonian. We conjecture that all cubes and higher-dimensional grids are stackable, and leave the reader with several open problems, questions, and generalizations

    Generation and properties of random graphs and analysis of randomized algorithms

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    We study a new method of generating random dd-regular graphs by repeatedly applying an operation called pegging. The pegging algorithm, which applies the pegging operation in each step, is a method of generating large random regular graphs beginning with small ones. We prove that the limiting joint distribution of the numbers of short cycles in the resulting graph is independent Poisson. We use the coupling method to bound the total variation distance between the joint distribution of short cycle counts and its limit and thereby show that O(ϵ−1)O(\epsilon^{-1}) is an upper bound of the \eps-mixing time. The coupling involves two different, though quite similar, Markov chains that are not time-homogeneous. We also show that the ϵ\epsilon-mixing time is not o(ϵ−1)o(\epsilon^{-1}). This demonstrates that the upper bound is essentially tight. We study also the connectivity of random dd-regular graphs generated by the pegging algorithm. We show that these graphs are asymptotically almost surely dd-connected for any even constant d≥4d\ge 4. The problem of orientation of random hypergraphs is motivated by the classical load balancing problem. Let h>w>0h>w>0 be two fixed integers. Let \orH be a hypergraph whose hyperedges are uniformly of size hh. To {\em ww-orient} a hyperedge, we assign exactly ww of its vertices positive signs with respect to this hyperedge, and the rest negative. A (w,k)(w,k)-orientation of \orH consists of a ww-orientation of all hyperedges of \orH, such that each vertex receives at most kk positive signs from its incident hyperedges. When kk is large enough, we determine the threshold of the existence of a (w,k)(w,k)-orientation of a random hypergraph. The (w,k)(w,k)-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The other topic we discuss is computing the probability of induced subgraphs in a random regular graph. Let 0<s<n0<s<n and HH be a graph on ss vertices. For any S⊂[n]S\subset [n] with ∣S∣=s|S|=s, we compute the probability that the subgraph of Gn,d\mathcal{G}_{n,d} induced by SS is HH. The result holds for any d=o(n1/3)d=o(n^{1/3}) and is further extended to Gn,d\mathcal{G}_{n,{\bf d}}, the probability space of random graphs with given degree sequence d\bf d. This result provides a basic tool for studying properties, for instance the existence or the counts, of certain types of induced subgraphs

    Extremal Results for Peg Solitaire on Graphs

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    In a 2011 paper by Beeler and Hoilman, the game of peg solitaire is generalized to arbitrary boards. These boards are treated as graphs in the combinatorial sense. An open problem from that paper is to determine the minimum number of edges necessary for a graph with a fixed number of vertices to be solvable. This thesis provides new bounds on this number. It also provides necessary and sufficient conditions for two families of graphs to be solvable, along with criticality results, and the maximum number of pegs that can be left in each of the two graph families

    Study of the cycle-to-cycle variations of an internal combustion engine fuelled with natural gas/hydrogen blends from the diagnosis of combustion pressure

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    Producción CientíficaSe presenta una metodología para el estudio de la influencia de la adicción de hidrógeno (de 0 a 100% en sustitución del gas natural en mezclas de gas natural/aire en la dispersión cíclica de la combustión de un motor de encendido provocado. Se utiliza un algoritmo genético que optimiza un modelo de diagnóstico de la combustión a partir de la presión medida experimentalmente en la cámara de combustión de forma que se optimizan entradas al modelo de diágnóstico como son el ángulo de la presión máxima y el offset de presión, la relación de compresión volumétrica, el índice de la transferencia de calor a las paredes, etc. Se observa que la adición de hidrógeno aumenta mucho la velocidad de combustión a partir de un valor del 60%.Ministerio de economía y competitividad de Españ

    Macroeconomic stabilisation and intervention policy under an exchange rate band

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    Exchange Rate;Stabilization;Foreign Exchange Market
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