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The Circular Chromatic Number of SeriesParallel Graphs
"... In this paper, we consider the circular chromatic number c (G) of seriesparallel graphs G. It is well known that seriesparallel graphs have chromatic number at most 3. Hence their circular chromatic number is also at most 3. If a seriesparallel graph G contains a triangle, then both the chro ..."
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Cited by 8 (4 self)
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the chromatic number and the circular chromatic number of G are indeed equal to 3. We shall show that if a seriesparallel graph G has girth at least 2b(3k \Gamma 1)=2c, then c (G) 4k=(2k \Gamma 1). The special case k = 2 of this result implies that a triangle free seriesparallel graph G has circular
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combin ..."
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Cited by 423 (37 self)
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sparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several wellknown signal and image processing problems can be cast as demanding solutions of undetermined systems
Circular Flow and Circular Chromatic Number . . .
, 2007
"... This thesis considers circular flowtype and circular chromatictype parameters (φ and χ, respectively) for matroids. In particular we focus on orientable matroids and 6√1matroids. These parameters are obtained via two approaches: algebraic and orientationbased. The general questions we discuss ..."
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This thesis considers circular flowtype and circular chromatictype parameters (φ and χ, respectively) for matroids. In particular we focus on orientable matroids and 6√1matroids. These parameters are obtained via two approaches: algebraic and orientationbased. The general questions we discuss
A note on generalized chromatic number and generalized girth
 Discrete Math
, 2000
"... Abstract. Erdős proved that there are graphs with arbitrarily large girth and chromatic number. We study the extension of this for generalized chromatic numbers. Generalized graph coloring describes the partitioning of the vertices into classes whose induced subgraphs satisfy particular constraints. ..."
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Cited by 3 (0 self)
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Abstract. Erdős proved that there are graphs with arbitrarily large girth and chromatic number. We study the extension of this for generalized chromatic numbers. Generalized graph coloring describes the partitioning of the vertices into classes whose induced subgraphs satisfy particular constraints
A New Proof of the Girth Chromatic Number Theorem
, 2004
"... We give a new proof of the classical Erdös theorem on the existence of graphs with arbitrarily high chromatic number and girth. Rather than considering random graphs where the edges are chosen with some carefully adjusted probability, we use a simple counting argument on a set of graphs with bounded ..."
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We give a new proof of the classical Erdös theorem on the existence of graphs with arbitrarily high chromatic number and girth. Rather than considering random graphs where the edges are chosen with some carefully adjusted probability, we use a simple counting argument on a set of graphs
The Chromatic Number of Oriented Graphs
 J. Graph Theory
, 2001
"... . We introduce in this paper the notion of the chromatic number of an oriented graph G (that is of an antisymmetric directed graph) dened as the minimum order of an oriented graph H such that G admits a homomorphism to H . We study the chromatic number of oriented ktrees and of oriented graphs with ..."
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Cited by 61 (20 self)
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. We introduce in this paper the notion of the chromatic number of an oriented graph G (that is of an antisymmetric directed graph) dened as the minimum order of an oriented graph H such that G admits a homomorphism to H . We study the chromatic number of oriented ktrees and of oriented graphs
Minors in Graphs of Large Girth
 J. Combin. Theory B
, 1988
"... We show that for every odd integer g 5 there exists a constant c such that every graph of minimum degree r and girth at least g contains a minor of minimum degree at least cr . This is best possible up to the value of the constant c for g = 5; 7 and 11. More generally, a wellknown conjecture ..."
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Cited by 5 (0 self)
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conjecture about the minimal order of graphs of given minimum degree and large girth would imply that our result gives the correct order of magnitude for all odd values of g. The case g = 5 of our result implies Hadwiger's conjecture for C 4 free graphs of suciently large chromatic number.
A Revival of the Girth Conjecture
, 2003
"... We show that for each " > 0, there exists a number g such that the circular chromatic index of every cubic bridgeless graph with girth at least g is at most 3 + ". This contrasts to the fact (which disproved the Girth Conjecture) that there are snarks of arbitrary large girth. In par ..."
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Cited by 7 (3 self)
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We show that for each " > 0, there exists a number g such that the circular chromatic index of every cubic bridgeless graph with girth at least g is at most 3 + ". This contrasts to the fact (which disproved the Girth Conjecture) that there are snarks of arbitrary large girth
Constructions for Cubic Graphs With Large Girth
 Electronic Journal of Combinatorics
, 1998
"... The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a welldefined integer ¯ 0 (g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value ¯ 0 (g) is attained by a ..."
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Cited by 53 (1 self)
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The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a welldefined integer ¯ 0 (g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value ¯ 0 (g) is attained
LARGEGIRTH ROOTS OF GRAPHS
, 2010
"... We study the problem of recognizing graph powers and computing roots of graphs. We provide a polynomial time recognition algorithm for rth powers of graphs of girth at least 2r + 3, thus improving a bound conjectured by Farzad et al. (STACS 2009). Our algorithm also finds all rth roots of a give ..."
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Cited by 1 (0 self)
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given graph that have girth at least 2r + 3 and no degree one vertices, which is a step towards a recent conjecture of Levenshtein that such root should be unique. On the negative side, we prove that recognition becomes an NPcomplete problem when the bound on girth is about twice smaller. Similar
Results 11  20
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4,705