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Constant C (constant + c)
Selected AbstractsSubgraph-avoiding coloring of graphsJOURNAL OF GRAPH THEORY, Issue 4 2010Jia Shen Abstract Given a "forbidden graph" F and an integer k, an F-avoiding k-coloring of a graph G is a k -coloring of the vertices of G such that no maximal F -free subgraph of G is monochromatic. The F-avoiding chromatic numberacF(G) is the smallest integer k such that G is F -avoiding k -colorable. In this paper, we will give a complete answer to the following question: for which graph F, does there exist a constant C, depending only on F, such that acF(G) , C for any graph G? For those graphs F with unbounded avoiding chromatic number, upper bounds for acF(G) in terms of various invariants of G are also given. Particularly, we prove that , where n is the order of G and F is not Kk or . © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 300,310, 2010 [source] Between ends and fibersJOURNAL OF GRAPH THEORY, Issue 2 2007C. Paul Bonnington Abstract Let , be an infinite, locally finite, connected graph with distance function ,. Given a ray P in , and a constant C , 1, a vertex-sequence is said to be regulated by C if, for all n,,, never precedes xn on P, each vertex of P appears at most C times in the sequence, and . R. Halin (Math. Ann., 157, 1964, 125,137) defined two rays to be end-equivalent if they are joined by infinitely many pairwise-disjoint paths; the resulting equivalence classes are called ends. More recently H. A. Jung (Graph Structure Theory, Contemporary Mathematics, 147, 1993, 477,484) defined rays P and Q to be b-equivalent if there exist sequences and VQ regulated by some constant C , 1 such that for all n,,; he named the resulting equivalence classes b-fibers. Let denote the set of nondecreasing functions from into the set of positive real numbers. The relation (called f-equivalence) generalizes Jung's condition to . As f runs through , uncountably many equivalence relations are produced on the set of rays that are no finer than b -equivalence while, under specified conditions, are no coarser than end-equivalence. Indeed, for every , there exists an "end-defining function" that is unbounded and sublinear and such that implies that P and Q are end-equivalent. Say if there exists a sublinear function such that . The equivalence classes with respect to are called bundles. We pursue the notion of "initially metric" rays in relation to bundles, and show that in any bundle either all or none of its rays are initially metric. Furthermore, initially metric rays in the same bundle are end-equivalent. In the case that , contains translatable rays we give some sufficient conditions for every f -equivalence class to contain uncountably many g -equivalence classes (where ). We conclude with a variety of applications to infinite planar graphs. Among these, it is shown that two rays whose union is the boundary of an infinite face of an almost-transitive planar map are never bundle- equivalent. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 125,153, 2007 [source] Approximating the smallest k -edge connected spanning subgraph by LP-roundingNETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2009Harold N. Gabow Abstract The smallest k-ECSS problem is, given a graph along with an integer k, find a spanning subgraph that is k -edge connected and contains the fewest possible number of edges. We examine a natural approximation algorithm based on rounding an LP solution. A tight bound on the approximation ratio is 1 + 3/k for undirected graphs with k > 1 odd, 1 + 2/k for undirected graphs with k even, and 1 + 2/k for directed graphs with k arbitrary. Using iterated rounding improves the first upper bound to 1 + 2/k. On the hardness side we show that for some absolute constant c > 0, for any integer k , 2 (k , 1), a polynomial-time algorithm approximating the smallest k -ECSS on undirected (directed) multigraphs to within ratio 1 + c/k would imply P = NP. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009 [source] Coloring H-free hypergraphsRANDOM STRUCTURES AND ALGORITHMS, Issue 1 2010Tom Bohman Abstract Fix r , 2 and a collection of r -uniform hypergraphs . What is the minimum number of edges in an -free r -uniform hypergraph with chromatic number greater than k? We investigate this question for various . Our results include the following: An (r,l)-system is an r -uniform hypergraph with every two edges sharing at most l vertices. For k sufficiently large, there is an (r,l)-system with chromatic number greater than k and number of edges at most c(kr,1 log k)l/(l,1), where This improves on the previous best bounds of Kostochka et al. (Random Structures Algorithms 19 (2001), 87,98). The upper bound is sharp apart from the constant c as shown in (Random Structures Algorithms 19 (2001) 87,98). The minimum number of edges in an r -uniform hypergraph with independent neighborhoods and chromatic number greater than k is of order kr+1/(r,1) log O(1)k as k , ,. This generalizes (aside from logarithmic factors) a result of Gimbel and Thomassen (Discrete Mathematics 219 (2000), 275,277) for triangle-free graphs. Let T be an r -uniform hypertree of t edges. Then every T -free r -uniform hypergraph has chromatic number at most 2(r , 1)(t , 1) + 1. This generalizes the well-known fact that every T -free graph has chromatic number at most t. Several open problems and conjectures are also posed. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010 [source] Minors in random regular graphsRANDOM STRUCTURES AND ALGORITHMS, Issue 4 2009Nikolaos Fountoulakis Abstract We show that there is a constant c so that for fixed r , 3 a.a.s. an r -regular graph on n vertices contains a complete graph on vertices as a minor. This confirms a conjecture of Markström (Ars Combinatoria 70 (2004) 289,295). Since any minor of an r -regular graph on n vertices has at most rn/2 edges, our bound is clearly best possible up to the value of the constant c. As a corollary, we also obtain the likely order of magnitude of the largest complete minor in a random graph Gn,p during the phase transition (i.e., when pn , 1). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009 [source] Distances in random graphs with finite variance degreesRANDOM STRUCTURES AND ALGORITHMS, Issue 1 2005Remco van der Hofstad In this paper we study a random graph with N nodes, where node j has degree Dj and {Dj} are i.i.d. with ,(Dj , x) = F(x). We assume that 1 , F(x) , cx,,+1 for some , > 3 and some constant c > 0. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when N , ,. We prove that the graph distance grows like log,N, when the base of the logarithm equals , = ,,[Dj(Dj , 1)]/,,[Dj] > 1. This confirms the heuristic argument of Newman, Strogatz, and Watts [Phys Rev E 64 (2002), 026118, 1,17]. In addition, the random fluctuations around this asymptotic mean log,N are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005 [source] PLUG-IN ESTIMATION OF GENERAL LEVEL SETSAUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 1 2006Antonio Cuevas Summary Given an unknown function (e.g. a probability density, a regression function, ,) f and a constant c, the problem of estimating the level set L(c) ={f,c} is considered. This problem is tackled in a very general framework, which allows f to be defined on a metric space different from . Such a degree of generality is motivated by practical considerations and, in fact, an example with astronomical data is analyzed where the domain of f is the unit sphere. A plug-in approach is followed; that is, L(c) is estimated by Ln(c) ={fn,c}, where fn is an estimator of f. Two results are obtained concerning consistency and convergence rates, with respect to the Hausdorff metric, of the boundaries ,Ln(c) towards ,L(c). Also, the consistency of Ln(c) to L(c) is shown, under mild conditions, with respect to the L1 distance. Special attention is paid to the particular case of spherical data. [source] The average-case area of Heilbronn-type triangles,RANDOM STRUCTURES AND ALGORITHMS, Issue 2 2002Tao Jiang From among triangles with vertices chosen from n points in the unit square, let T be the one with the smallest area, and let A be the area of T. Heilbronn's triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the average-case: If the n points are chosen independently and at random (with a uniform distribution), then there exist positive constants c and C such that c/n3<,n
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