Cliques in Graphs Excluding a Complete Graph Minor

Wood, David R.

doi:10.37236/5715

Cited by 14 publications

(17 citation statements)

References 35 publications

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“…In some sense, degeneracy is equivalent with "sparsity." For example, the number of cliques in every r-degenerate graph is at most a linear number of its vertices [50]. On the other hand, a graph is not r-degenerate if and only if it contains a subgraph of minimum degree at least r + 1.…”

Section: Introductionmentioning

confidence: 99%

Phase Transition of Degeneracy in Minor-Closed Families

Liu

Fan

2019

Preprint

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Given an infinite family G of graphs and a monotone property P, an (upper) threshold for G and P is a "fastest growing" function p :is the random subgraph of G n such that each edge remains independently with probability p(n).In this paper we study the upper threshold for the family of H-minor free graphs and for the graph property of being (r − 1)-degenerate, which is one fundamental graph property that has been shown widely applicable to various problems in graph theory. Even a constant factor approximation for the upper threshold for all pairs (r, H) is expected to be very difficult by its close connection to a major open question in extremal graph theory. We determine asymptotically the thresholds (up to a constant factor) for being (r − 1)-degenerate for a large class of pairs (r, H), including all graphs H of minimum degree at least r and all graphs H with no vertex-cover of size at most r, and provide lower bounds for the rest of the pairs of (r, H). The results generalize to arbitrary proper minor-closed families and the properties of being r-colorable, being r-choosable, or containing an r-regular subgraph, respectively.

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Section: Introductionmentioning

confidence: 99%

Phase Transition of Degeneracy in Minor-Closed Families

Liu

Fan

2019

Preprint

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“…The average size of the cliques in the complement of a perfect matching of size x is 2x/3, and a random clique in this graph typically has about this size. Thus, for k = 4t/9, the graph which is a complement of a perfect matching of size just less than 2t/3 is K t -minor free and has nearly the maximum number of K k , namely 3 2t/3−o(t) n. For k = t − 1 and n ≥ t − 1, Wood [18] shows that the maximum number of K k is n − t + 2. We therefore see that depending on the range of k, the answer has quite different forms.…”

Section: Discussionmentioning

confidence: 96%

“…We determined the number of cliques in a graph on n vertices with no K t -minor up to a factor 2 o(t) . It would be interesting to determine the exact value, as has been done by Wood [18] for t ≤ 9. As observed by Wood, for small values of t, the extremal example is a graph formed from a K t−2 by adding vertices one at a time whose neighorhood in the graph so far is a K t−2 .…”

Section: Discussionmentioning

confidence: 99%

“…They proved that every graph on n vertices with no K t -subdivision (and hence every graph on n vertices with no K t -minor) has at most 2 5t+o(t) n cliques, and observed that optimizing their proof would improve the exponential constant from 5 to a number less than 4. Wood [18] very recently determined, for t ≤ 9 and n ≥ t − 2, that the maximum number of cliques a K t -minor free graph on n vertices can have is 2 t−2 (n − t + 3). He further conjectures that this bound holds if and only if t ≤ 49.…”

Section: Introductionmentioning

confidence: 99%

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On the number of cliques in graphs with a forbidden minor

Fox

Fan

2017

Journal of Combinatorial Theory, Series B

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Reed and Wood and independently Norine, Seymour, Thomas, and Wollan proved that for each positive integer t there is a constant c(t) such that every graph on n vertices with no Kt-minor has at most c(t)n cliques. Wood asked in 2007 if we can take c(t) = c t for some absolute constant c.This question was recently answered affirmatively by Lee and Oum. In this paper, we determine the exponential constant. We prove that every graph on n vertices with no Kt-minor has at most 3 2t/3+o(t) n cliques. This bound is tight for n ≥ 4t/3. More generally, let H be a connected graph on t vertices, and x denote the size (i.e., the number edges) of the largest matching in the complement of H. We prove that every graph on n vertices with no H-minor has at most max(3 2t/3−x/3+o(t) n, 2 t+o(t) n) cliques, and this bound is tight for n ≥ max(4t/3 − 2x/3, t) by a simple construction. Even more generally, we determine explicitly the exponential constant for the maximum number of cliques an n-vertex graph can have in a minor-closed family of graphs which is closed under disjoint union.

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“…As a consequence, H is dek -degenerate. It is a folklore result that -degenerate graphs on n vertices have at most t−1 n cliques of size t (see for instance [28,Lemma 18], where the proof gives a linear time algorithm to enumerate all the cliques of size t when t and are fixed), and hence there are at most…”

Section: Lemmamentioning

confidence: 99%

Packing and Covering Balls in Graphs Excluding a Minor

et al. 2021

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We prove that for every integer t 1 there exists a constant ct such that for every Kt-minor-free graph G, and every set S of balls in G, the minimum size of a set of vertices of G intersecting all the balls of S is at most ct times the maximum number of vertex-disjoint balls in S. This was conjectured by Chepoi, Estellon, and Vaxès in 2007 in the special case of planar graphs and of balls having the same radius. N. Bousquet is supported by ANR Project DISTANCIA (ANR-17-CE40-0015). W. Cames van Batenburg, G. Joret, and F. Pirot are supported by an ARC grant from the Wallonia-Brussels Federation of Belgium. L. Esperet is supported by ANR Projects GATO (ANR-16-CE40-0009-01) and GrR (ANR-18-CE40-0032). C.

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Cliques in Graphs Excluding a Complete Graph Minor

Cited by 14 publications

References 35 publications

Phase Transition of Degeneracy in Minor-Closed Families

Phase Transition of Degeneracy in Minor-Closed Families

On the number of cliques in graphs with a forbidden minor

Packing and Covering Balls in Graphs Excluding a Minor

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