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Asymptotic enumeration and limit laws of seriesparallel graphs, manuscript in preparation
"... Abstract. We show that the number gn of labelled seriesparallel graphs on n vertices is asymptotically gn ∼ g · n −5/2 γ n n!, where γ and g are explicit computable constants. We show that the number of edges in random seriesparallel graphs is asymptotically normal with linear mean and variance, an ..."
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Cited by 16 (11 self)
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Abstract. We show that the number gn of labelled seriesparallel graphs on n vertices is asymptotically gn ∼ g · n −5/2 γ n n!, where γ and g are explicit computable constants. We show that the number of edges in random seriesparallel graphs is asymptotically normal with linear mean and variance, and that the number of edges is sharply concentrated around its expected value. Similar results are proved for labelled outerplanar graphs and for graphs not containing K2,3 as a minor. 1.
On the number of series parallel and outerplanar graphs
 In EuroComb ’05, volume AE of DMTCS Proceedings
, 2005
"... We show that the number gn of labelled seriesparallel graphs on n vertices is asymptotically gn ∼ g · n −5/2 γ n n!, where γ and g are explicit computable constants. We show that the number of edges in random seriesparallel graphs is asymptotically normal with linear mean and variance, and that th ..."
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Cited by 8 (3 self)
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We show that the number gn of labelled seriesparallel graphs on n vertices is asymptotically gn ∼ g · n −5/2 γ n n!, where γ and g are explicit computable constants. We show that the number of edges in random seriesparallel graphs is asymptotically normal with linear mean and variance, and that the number of edges is sharply concentrated around its expected value. Similar results are proved for labelled outerplanar graphs.
Linear choosability of graphs
"... A proper vertex coloring of a non oriented graph G = (V, E) is linear if the graph induced by the vertices of two color classes is a forest of paths. A graph G is Llist colorable if for a given list assignment L = {L(v) : v ∈ V}, there exists a proper coloring c of G such that c(v) ∈ L(v) for all ..."
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Cited by 4 (0 self)
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A proper vertex coloring of a non oriented graph G = (V, E) is linear if the graph induced by the vertices of two color classes is a forest of paths. A graph G is Llist colorable if for a given list assignment L = {L(v) : v ∈ V}, there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V. If G is Llist colorable for every list assignment with L(v)  ≥ k for all v ∈ V, then G is said kchoosable. A graph is said to be lineary kchoosable if the coloring obtained is linear. In this paper, we investigate the linear choosability of graphs for some families of graphs: graphs with small maximum degree, with given maximum average degree, planar graphs... Moreover, we prove that determining whether a bipartite subcubic planar graph is lineary 3colorable is an NPcomplete problem.
Random Cubic Planar Graphs
"... We show that the number of labeled cubic planar graphs on n vertices with n even is asymptotically αn −7/2 ρ −n n!, where ρ −1. = 3.13259 and α are analytic constants. We show also that the chromatic number of a random cubic planar graph that is chosen uniformly at random among all the labeled cubic ..."
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Cited by 2 (1 self)
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We show that the number of labeled cubic planar graphs on n vertices with n even is asymptotically αn −7/2 ρ −n n!, where ρ −1. = 3.13259 and α are analytic constants. We show also that the chromatic number of a random cubic planar graph that is chosen uniformly at random among all the labeled cubic planar graphs on n vertices is three with probability tending to e −ρ4 /4!. = 0.999568, and is four with probability tending to 1−e −ρ 4 /4! as n → ∞ with n even. The proof given combines generating function techniques with probabilistic arguments.
3.3. Theoretical Validation 7
"... c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1 ..."
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c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1
3.3. Theoretical Validation 6
"... c t i v it y e p o r t 2008 Table of contents 1. Team.................................................................................... 1 ..."
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c t i v it y e p o r t 2008 Table of contents 1. Team.................................................................................... 1
Generating Unlabeled Connected Cubic Planar Graphs Uniformly at Random
"... Abstract. We present an expected polynomial time algorithm to generate an unlabeled connected cubic planar graph uniformly at random. We first consider rooted connected cubic planar graphs, i.e., we count connected cubic planar graphs up to isomorphisms that fix a certain directed edge. Based on dec ..."
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Abstract. We present an expected polynomial time algorithm to generate an unlabeled connected cubic planar graph uniformly at random. We first consider rooted connected cubic planar graphs, i.e., we count connected cubic planar graphs up to isomorphisms that fix a certain directed edge. Based on decompositions along the connectivity structure, we derive recurrence formulas for the exact number of rooted cubic planar graphs. This leads to rooted 3connected cubic planar graphs, which have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sensereversing automorphism. Therefore we introduce the concept of colored networks, which stand in bijective correspondence to rooted 3connected cubic planar graphs with given symmetries. Colored networks can again be decomposed along the connectivity structure. For rooted 3connected cubic planar graphs embedded in the plane, we switch to the dual and count rooted triangulations. Since all these numbers can be evaluated in polynomial time using dynamic programming, rooted connected cubic planar graphs can be generated uniformly at random in polynomial time by inverting the decomposition along the connectivity structure. To generate connected cubic planar graphs without a root uniformly at random, we apply rejection sampling and obtain an expected polynomial time algorithm. 1