Random unlabelled graphs containing few disjoint cycles

Kang, Mihyun; McDiarmid, Colin

doi:10.1002/rsa.20355

Cited by 6 publications

(7 citation statements)

References 11 publications

(27 reference statements)

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“…Thus Ex (k+1)C consists of apex k F together with an exponentially smaller class of 'exceptional' graphs. A similar result holds for unlabelled graphs [8]; we consider only labelled graphs in this paper. The Erdős-Pósa theorem was generalised in 1986 by Robertson and Seymour [17].…”

Section: Introductionsupporting

confidence: 66%

“…To be more precise, it was shown in that as

n \to \infty

| {(Ex (k + 1) C)}_{n} | = (1 + e^{- Ω (n)}) | {({apex}^{k} ℱ)}_{n} | .

Thus

Ex (k + 1) C

consists of

{apex}^{k} ℱ

together with an exponentially smaller class of ‘exceptional’ graphs. A similar result holds for unlabelled graphs ; we consider only labelled graphs in this paper.…”

Section: Introductionsupporting

confidence: 60%

“…where R = R(A, ρ/2 k ) is the Boltzmann Poisson random graph for A and ρ/2 k as in (8) and (9) above. Further,…”

Section: Properties Of the Random Graphs R Nmentioning

confidence: 99%

“…Then the exponential generating function B(x) of B satisfies 0 < B(ρ) < ∞; and d TV (UFrag(R n ), R) → 0 where R = R(B, ρ) is the Boltzmann Poisson random graph for B and ρ as defined in (8) and (9) Proof of (11) in Theorem 6.1. Let A k+1 denote Ex (k + 1)B.…”

Section: Random Structures and Algorithms Doi 101002/rsamentioning

confidence: 99%

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Random graphs containing few disjoint excluded minors

McDiarmid

Kurauskas

2012

Random Struct Algorithms

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The Erdős‐Pósa theorem (1965) states that in each graph G which contains at most k disjoint cycles, there is a ‘blocking’ set B of at most f(k) vertices such that the graph G – B is acyclic. Robertson and Seymour (1986) give an extension concerning any minor‐closed class scriptA of graphs, as long as scriptA does not contain all planar graphs: in each graph G which contains at most k disjoint excluded minors for scriptA, there is a set B of at most g(k) vertices such that G – B is in scriptA. In an earlier paper (Kurauskas and McDiarmid, Combin, Probab Comput 20 (2011) 763–775), we showed that, amongst all graphs on vertex set [n]={1,…,n} which contain at most k disjoint cycles, all but an exponentially small proportion contain a blocking set of just k vertices. In the present paper we build on the previous work, and give an extension concerning any minor‐closed graph class scriptA with 2‐connected excluded minors, as long as scriptA does not contain all fans (here a ‘fan’ is a graph consisting of a path together with a vertex joined to each vertex on the path). We show that amongst all graphs G on [n] which contain at most k disjoint excluded minors for scriptA, all but an exponentially small proportion contain a set B of k vertices such that G – B is in scriptA. (This is not the case when scriptA contains all fans.) For a random graph Rn sampled uniformly from the graphs on [n] with at most k disjoint excluded minors for scriptA, we consider also vertex degrees and the uniqueness of small blockers, the clique number and chromatic number, and the probability of being connected. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 240‐268, 2014

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Section: Introductionsupporting

confidence: 66%

“…To be more precise, it was shown in that as

n \to \infty

| {(Ex (k + 1) C)}_{n} | = (1 + e^{- Ω (n)}) | {({apex}^{k} ℱ)}_{n} | .

Thus

Ex (k + 1) C

consists of

{apex}^{k} ℱ

together with an exponentially smaller class of ‘exceptional’ graphs. A similar result holds for unlabelled graphs ; we consider only labelled graphs in this paper.…”

Section: Introductionsupporting

confidence: 60%

“…where R = R(A, ρ/2 k ) is the Boltzmann Poisson random graph for A and ρ/2 k as in (8) and (9) above. Further,…”

Section: Properties Of the Random Graphs R Nmentioning

confidence: 99%

Section: Random Structures and Algorithms Doi 101002/rsamentioning

confidence: 99%

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Random graphs containing few disjoint excluded minors

McDiarmid

Kurauskas

2012

Random Struct Algorithms

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“…Analytic work in papers such as those just mentioned much extends earlier combinatorial and probabilistic investigations, as for example in [33,34,35,48,49,52,53,54]. For further recent related work (appearing in 2010 or later) see for example [6,10,16,20,21,22,23,24,25,26,30,42,43,44,45,50,57,58].…”

Section: Introductionmentioning

confidence: 79%

Random Graphs from a Weighted Minor-Closed Class

McDiarmid

2013

Electron. J. Combin.

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There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable minor-closed class, such as the class of all planar graphs. Here we use combinatorial and probabilistic methods to investigate a more general model. We consider random graphs from a 'well-behaved' class of graphs: examples of such classes include all minor-closed classes of graphs with 2connected excluded minors (such as forests, series-parallel graphs and planar graphs), the class of graphs embeddable on any given surface, and the class of graphs with at most k vertex-disjoint cycles. Also, we give weights to edges and components to specify probabilities, so that our random graphs correspond to the random cluster model, appropriately conditioned.We find that earlier results extend naturally in both directions, to general well-behaved classes of graphs, and to the weighted framework, for example results concerning the probability of a random graph being connected; and we also give results on the 2-core which are new even for the uniform (unweighted) case.

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On graphs with few disjoint t-star minors

McDiarmid¹

2011

European Journal of Combinatorics

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Random unlabelled graphs containing few disjoint cycles

Cited by 6 publications

References 11 publications

Random graphs containing few disjoint excluded minors

Random graphs containing few disjoint excluded minors

Random Graphs from a Weighted Minor-Closed Class

On graphs with few disjoint t-star minors

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