47,284 research outputs found

    Graphs with few spanning substructures

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    In this thesis, we investigate a number of problems related to spanning substructures of graphs. The first few chapters consider extremal problems related to the number of forest-like structures of a graph. We prove that one can find a threshold graph which contains the minimum number of spanning pseudoforests, as well as rooted spanning forests, amongst all graphs on n vertices and e edges. This has left the open question of exactly which threshold graphs have the minimum number of these spanning substructures. We make progress towards this question in particular cases of spanning pseudoforests. The final chapter takes on a different flavor---we determine the complexity of a problem related to Hamilton cycles in hypergraphs. Dirac\u27s theorem states that graphs with minimum degree at least half the size of the vertex set are guaranteed to have a Hamilton cycle. In 1993, Karpinksi, Dahlhaus, and Hajnal proved that for any c\u3c1/2, the problem of determining whether a graph with minimum degree at least cn has a Hamilton cycle is NP-complete. The analogous problem in hypergraphs, for both a Dirac-type condition and complexity, are just as interesting. We prove that for classes of hypergraphs with certain minimum vertex degree conditions, the problem of determining whether or not they contain an l-Hamilton cycle is NP-complete. Advisor: Professor Jamie Radcliff

    Feynman graph polynomials

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    The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson's identity and matroids.Comment: 35 pages, references adde

    Reconstructing pedigrees: some identifiability questions for a recombination-mutation model

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    Pedigrees are directed acyclic graphs that represent ancestral relationships between individuals in a population. Based on a schematic recombination process, we describe two simple Markov models for sequences evolving on pedigrees - Model R (recombinations without mutations) and Model RM (recombinations with mutations). For these models, we ask an identifiability question: is it possible to construct a pedigree from the joint probability distribution of extant sequences? We present partial identifiability results for general pedigrees: we show that when the crossover probabilities are sufficiently small, certain spanning subgraph sequences can be counted from the joint distribution of extant sequences. We demonstrate how pedigrees that earlier seemed difficult to distinguish are distinguished by counting their spanning subgraph sequences.Comment: 40 pages, 9 figure

    Edge Partitions of Optimal 22-plane and 33-plane Graphs

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    A topological graph is a graph drawn in the plane. A topological graph is kk-plane, k>0k>0, if each edge is crossed at most kk times. We study the problem of partitioning the edges of a kk-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for k=1k=1, we focus on optimal 22-plane and 33-plane graphs, which are 22-plane and 33-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple optimal 22-plane graph into a 11-plane graph and a forest, while (ii) an edge partition formed by a 11-plane graph and two plane forests always exists and can be computed in linear time. (iii) We describe efficient algorithms to partition the edges of a simple optimal 22-plane graph into a 11-plane graph and a plane graph with maximum vertex degree 1212, or with maximum vertex degree 88 if the optimal 22-plane graph is such that its crossing-free edges form a graph with no separating triangles. (iv) We exhibit an infinite family of simple optimal 22-plane graphs such that in any edge partition composed of a 11-plane graph and a plane graph, the plane graph has maximum vertex degree at least 66 and the 11-plane graph has maximum vertex degree at least 1212. (v) We show that every optimal 33-plane graph whose crossing-free edges form a biconnected graph can be decomposed, in linear time, into a 22-plane graph and two plane forests

    Optimal path and cycle decompositions of dense quasirandom graphs

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    Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let 0<p<10<p<1 be constant and let GGn,pG\sim G_{n,p}. Let odd(G)odd(G) be the number of odd degree vertices in GG. Then a.a.s. the following hold: (i) GG can be decomposed into Δ(G)/2\lfloor\Delta(G)/2\rfloor cycles and a matching of size odd(G)/2odd(G)/2. (ii) GG can be decomposed into max{odd(G)/2,Δ(G)/2}\max\{odd(G)/2,\lceil\Delta(G)/2\rceil\} paths. (iii) GG can be decomposed into Δ(G)/2\lceil\Delta(G)/2\rceil linear forests. Each of these bounds is best possible. We actually derive (i)--(iii) from `quasirandom' versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte

    On globally sparse Ramsey graphs

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    We say that a graph GG has the Ramsey property w.r.t.\ some graph FF and some integer r2r\geq 2, or GG is (F,r)(F,r)-Ramsey for short, if any rr-coloring of the edges of GG contains a monochromatic copy of FF. R{\"o}dl and Ruci{\'n}ski asked how globally sparse (F,r)(F,r)-Ramsey graphs GG can possibly be, where the density of GG is measured by the subgraph HGH\subseteq G with the highest average degree. So far, this so-called Ramsey density is known only for cliques and some trivial graphs FF. In this work we determine the Ramsey density up to some small error terms for several cases when FF is a complete bipartite graph, a cycle or a path, and r2r\geq 2 colors are available

    On the limiting distribution of the metric dimension for random forests

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    The metric dimension of a graph G is the minimum size of a subset S of vertices of G such that all other vertices are uniquely determined by their distances to the vertices in S. In this paper we investigate the metric dimension for two different models of random forests, in each case obtaining normal limit distributions for this parameter.Comment: 22 pages, 5 figure

    The looping rate and sandpile density of planar graphs

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    We give a simple formula for the looping rate of loop-erased random walk on a finite planar graph. The looping rate is closely related to the expected amount of sand in a recurrent sandpile on the graph. The looping rate formula is well-suited to taking limits where the graph tends to an infinite lattice, and we use it to give an elementary derivation of the (previously computed) looping rate and sandpile densities of the square, triangular, and honeycomb lattices, and compute (for the first time) the looping rate and sandpile densities of many other lattices, such as the kagome lattice, the dice lattice, and the truncated hexagonal lattice (for which the values are all rational), and the square-octagon lattice (for which it is transcendental)
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