45,949 research outputs found

    An algebraic formulation of the graph reconstruction conjecture

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    The graph reconstruction conjecture asserts that every finite simple graph on at least three vertices can be reconstructed up to isomorphism from its deck - the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important tool in graph reconstruction. Roughly speaking, given the deck of a graph GG and any finite sequence of graphs, it gives a linear constraint that every reconstruction of GG must satisfy. Let ψ(n)\psi(n) be the number of distinct (mutually non-isomorphic) graphs on nn vertices, and let d(n)d(n) be the number of distinct decks that can be constructed from these graphs. Then the difference ψ(n)d(n)\psi(n) - d(n) measures how many graphs cannot be reconstructed from their decks. In particular, the graph reconstruction conjecture is true for nn-vertex graphs if and only if ψ(n)=d(n)\psi(n) = d(n). We give a framework based on Kocay's lemma to study this discrepancy. We prove that if MM is a matrix of covering numbers of graphs by sequences of graphs, then d(n)rankR(M)d(n) \geq \mathsf{rank}_\mathbb{R}(M). In particular, all nn-vertex graphs are reconstructible if one such matrix has rank ψ(n)\psi(n). To complement this result, we prove that it is possible to choose a family of sequences of graphs such that the corresponding matrix MM of covering numbers satisfies d(n)=rankR(M)d(n) = \mathsf{rank}_\mathbb{R}(M).Comment: 12 pages, 2 figure

    The Graph Structure of Chebyshev Polynomials over Finite Fields and Applications

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    We completely describe the functional graph associated to iterations of Chebyshev polynomials over finite fields. Then, we use our structural results to obtain estimates for the average rho length, average number of connected components and the expected value for the period and preperiod of iterating Chebyshev polynomials

    Embedding bounded degree spanning trees in random graphs

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    We prove that if a tree TT has nn vertices and maximum degree at most Δ\Delta, then a copy of TT can almost surely be found in the random graph G(n,Δlog5n/n)\mathcal{G}(n,\Delta\log^5 n/n).Comment: 14 page

    Partitions and Coverings of Trees by Bounded-Degree Subtrees

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    This paper addresses the following questions for a given tree TT and integer d2d\geq2: (1) What is the minimum number of degree-dd subtrees that partition E(T)E(T)? (2) What is the minimum number of degree-dd subtrees that cover E(T)E(T)? We answer the first question by providing an explicit formula for the minimum number of subtrees, and we describe a linear time algorithm that finds the corresponding partition. For the second question, we present a polynomial time algorithm that computes a minimum covering. We then establish a tight bound on the number of subtrees in coverings of trees with given maximum degree and pathwidth. Our results show that pathwidth is the right parameter to consider when studying coverings of trees by degree-3 subtrees. We briefly consider coverings of general graphs by connected subgraphs of bounded degree

    The Cover Pebbling Number of Graphs

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    A pebbling move on a graph consists of taking two pebbles off of one vertex and placing one pebble on an adjacent vertex. In the traditional pebbling problem we try to reach a specified vertex of the graph by a sequence of pebbling moves. In this paper we investigate the case when every vertex of the graph must end up with at least one pebble after a series of pebbling moves. The cover pebbling number of a graph is the minimum number of pebbles such that however the pebbles are initially placed on the vertices of the graph we can eventually put a pebble on every vertex simultaneously. We find the cover pebbling numbers of trees and some other graphs. We also consider the more general problem where (possibly different) given numbers of pebbles are required for the vertices.Comment: 12 pages. Submitted to Discrete Mathematic
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