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AIexander Shen 《Mathematical Intelligencer》2000,22(3):20-21
This column is devoted to mathematics for fun. What better purpose is there for mathematics? To appear here, a theorem or problem or remark does not need to be profound (but it is allowed to be); it may not be directed only at specialists; it must attract and fascinate. We welcome, encourage, and frequently publish contributions from readers—either new notes, or replies to past columns. 相似文献
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Crista Arangala 《Complexity》2016,21(5):155-161
The solution to the game of Lights Out and its many variants have been well studied. This article introduces Langton's turmite to the game of Lights Out and discusses when Langton's turmite can solve Lights Out on a torus. The article also explores the behavior of multiple Langton's turmites on a Lights Out on a torus. © 2014 Wiley Periodicals, Inc. Complexity 21: 155–161, 2016 相似文献
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Paulo Ventura Araújo 《Elemente der Mathematik》2000,55(4):135-141
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W. R. Pulleyblank 《Journal of Graph Theory》1979,3(3):309-310
We show that the problem raised by Boesch, Suffel, and Tindell of determining whether or not a graph is spanned by an Eulerian subgraph is NP-complete. We also note that there does exist a good algorithm for determining if a graph is spanned by a subgraph having positive even degree at every node. 相似文献
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Shahar Golan 《Applied mathematics and computation》2011,217(14):6616-6623
Minesweeper is a popular single player game. It has been shown that the Minesweeper consistency problem is NP-complete and the Minesweeper counting problem is #P-complete. In this paper, we present efficient algorithms for solving these problems for Minesweeper graphs with bounded treewidth. Our algorithms turn out to be much better than those based directly on dynamic programming. The algorithms mostly use of algebraic operations on multivariate polynomials, so that one may use existing software to implement them easily. 相似文献
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Let G be a simple graph with adjacency matrix A, and p(x) a polynomial with rational coefficients. If p(A) is the adjacency matrix of a graph, we denote that graph by p(G). We consider the question: Given a graph G, which polynomials p(x) give rise to a graph p(G) and what are those graphs? We give a complete answer if G is a distance-regular graph. We then derive some general relations between the polynomials p(x), the spectrum of A, and the automorphism group of G. 相似文献
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§1.IntroductionLetGbeafinitegroupandSasubsetofGnotcontainingtheidentityelement1ofG.WedefinetheCayley(di)graphX=Cay(G,S)ofGwit... 相似文献
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The k-domination problem is to select a minimum cardinality vertex set D of a graph G such that every vertex of G is within distance k from some vertex of D. We consider a generalization of the k-domination problem, called the R-domination problem. A linear algorithm is presented that solves this problem for block graphs. Our algorithm is a generalization of Slater's algorithm [12], which is applicable for forest graphs. 相似文献
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Noga Alon Konstantin Makarychev Yury Makarychev Assaf Naor 《Inventiones Mathematicae》2006,163(3):499-522
We introduce a new graph parameter, called the Grothendieck constant of a graph G=(V,E), which is defined as the least constant K such that for every A:E→ℝ,
The classical Grothendieck inequality corresponds to the case of bipartite graphs, but the case of general graphs is shown to have various algorithmic applications. Indeed, our work is motivated by
the algorithmic problem of maximizing the quadratic form ∑{u,v}∈EA(u,v)ϕ(u)ϕ(v) over all ϕ:V→{-1,1}, which arises in the study of correlation clustering and in the investigation of the spin glass model. We give upper
and lower estimates for the integrality gap of this program. We show that the integrality gap is
, where
is the Lovász Theta Function of the complement of G, which is always smaller than the chromatic number ofG. This yields an efficient constant factor approximation algorithm for the above maximization problem for a wide range of
graphs G. We also show that the maximum possible integrality gap is always at least Ω(log ω(G)), where ω(G) is the clique number of G.
In particular it follows that the maximum possible integrality gap for the complete graph on n vertices with no loops is Θ(logn). More generally, the maximum possible integrality gap for any perfect graph with chromatic number n is Θ(logn). The lower bound for the complete graph improves a result of Kashin and Szarek on Gram matrices of uniformly bounded functions,
and settles a problem of Megretski and of Charikar and Wirth. 相似文献
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Krzysztof R. Apt Bart de Keijzer Mona Rahn Guido Schäfer Sunil Simon 《International Journal of Game Theory》2017,46(3):851-877
We introduce natural strategic games on graphs, which capture the idea of coordination in a local setting. We study the existence of equilibria that are resilient to coalitional deviations of unbounded and bounded size (i.e., strong equilibria and k-equilibria respectively). We show that pure Nash equilibria and 2-equilibria exist, and give an example in which no 3-equilibrium exists. Moreover, we prove that strong equilibria exist for various special cases. We also study the price of anarchy (PoA) and price of stability (PoS) for these solution concepts. We show that the PoS for strong equilibria is 1 in almost all of the special cases for which we have proven strong equilibria to exist. The PoA for pure Nash equilbria turns out to be unbounded, even when we fix the graph on which the coordination game is to be played. For the PoA for k-equilibria, we show that the price of anarchy is between \(2(n-1)/(k-1) - 1\) and \(2(n-1)/(k-1)\). The latter upper bound is tight for \(k=n\) (i.e., strong equilibria). Finally, we consider the problems of computing strong equilibria and of determining whether a joint strategy is a k-equilibrium or strong equilibrium. We prove that, given a coordination game, a joint strategy s, and a number k as input, it is co-NP complete to determine whether s is a k-equilibrium. On the positive side, we give polynomial time algorithms to compute strong equilibria for various special cases. 相似文献
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We study the behaviour of subharmonic functions on a graph. We assume bounds on the growth of balls and functions in order
to obtain Liouville type theorems. 相似文献
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Maria Chudnovsky Neil Robertson P. D. Seymour Robin Thomas 《Mathematical Programming》2003,97(1-2):405-422
A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect
graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can
be solved efficiently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear
programs always have an integer solution can be answered in terms of perfection of an associated graph.
In the first part of the paper we survey the main aspects of perfect graphs and their relevance. In the second part we outline
our recent proof of the Strong Perfect Graph Conjecture of Berge from 1961, the following: a graph is perfect if and only
if it has no induced subgraph isomorphic to an odd cycle of length at least five, or the complement of such an odd cycle.
Received: December 19, 2002 / Accepted: April 29, 2003
Published online: May 28, 2003
Key words. Berge graph – perfect graph – skew partition
Mathematics Subject Classification (1991): 05C17 相似文献
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