Dense Minors In Graphs Of Large Girth

Diestel, Reinhard; Rempel, Christof

doi:10.1007/s00493-005-0009-3

Cited by 21 publications

(18 citation statements)

References 11 publications

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“…They proved in [22] that for every odd integer g ≥ 5 there exists a constant c = c(g) > 0 such that every graph G of average degree r and without cycles shorter than g (such a graph is said to have girth more than g) contains a minor with average degree at least cr (g+1)/4 . This result improves significantly a much earlier result of Thomassen [37] and a recently obtained result by Diestel and Rompel [11]. Observe that the assumption for the case g = 5 essentially amounts to forbidding a 4-cycle, or K 2,2 ; thus this result of Kühn and Osthus establishes their above mentioned conjecture for the case s = s ′ = 2.…”

Section: Extremal Problems For Minorssupporting

confidence: 76%

Minors in Expanding Graphs

Krivelevich

Sudakov

2009

Geom. Funct. Anal.

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1 corresponding vertices u i , u j of Γ are connected by an edge. A graph G = (V, E) is (t, α)-expanding if every subset X ⊂ V of size |X| ≤ α|V |/t has at least t|X| external neighbors in G. A graphfor every subset X ⊆ V , where e(X) stands for the number of edges spanned by X in G. Informally, this definition indicates that the edge distribution of G is similar to that of the random graph G |V |,p , where the degree of similarity is controlled by parameter β.Here are the main results of this paper.Theorem 1 Let 0 < α < 1 be a constant. Let G be a (t, α)-expanding graph of order n, and let t ≥ 10. Then G contains a minor with average degree at least Theorem 2 Let G be a (p, β)-jumbled graph of order n such that β = o(np). Then G contains a minor with average degree cn √ p, for an absolute constant c > 0.This statement is an extension of results of A. Thomason [39,40], who studied the case of constant p. It can be also used to derive some of the results of Drier and Linial [12].Theorem 3 Let 2 ≤ s ≤ s ′ be integers. Let G be a K s,s ′ -free graph with average degree r. Then G contains a minor with average degree cr Theorem 4 Let k ≥ 2 and let G be a C 2k -free graph with average degree r. Then G contains a minor with average degree cr This theorem generalizes results of Thomassen [37], Diestel and Rompel [11], and Kühn and Osthus [22], who proved similar statements under the (much more restrictive) assumption that G has girth at least 2k + 1.All of the above results are, up to a constant factor, asymptotically tight (Theorems 1, 2), or are allegedly tight (Theorems 3, 4), where in the latter case the tightness hinges upon widely accepted conjectures from Extremal Graph Theory about the asymptotic behavior of the Turán numbers of K s,s ′ and of C 2k .2

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Section: Extremal Problems For Minorssupporting

confidence: 76%

Minors in Expanding Graphs

Krivelevich

Sudakov

2009

Geom. Funct. Anal.

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“…This fact was first observed by Thomassen [19], who obtained a bound on the girth linear in t. Diestel and Rempel [7] reduced it to 6 log 2 t ϩ 4. Theorem 1 applied with r ϭ 3 and k ϭ log 2 t ϩ 5 shows that the constant 6 can be reduced to 4.…”

Section: Corollarymentioning

confidence: 75%

“…As already observed in [7], the existence of 3-regular graphs of girth at least g and order at most c2 g/ 2 (which is a special case of the conjecture mentioned earlier) would show that Corollary 5 is asymptotically best possible in the sense that the constant 4 in the leading terms cannot be reduced any further (see Section 4). The minimal order of such 3-regular graphs is known to lie between c 1 2 g/ 2 and c 2 2 3g/4 .…”

Section: Corollarymentioning

confidence: 84%

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Minors in graphs of large girth

Kühn

Osthus

2003

Random Struct Algorithms

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ABSTRACT:We show that for every odd integer g Ն 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 ( gϩ1)/4 . This is best possible up to the value of the constant c for g ϭ 5, 7, and 11. More generally, a well-known 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 ϭ 5 of our result implies Hadwiger's conjecture for C 4 -free graphs of sufficiently large chromatic number.

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“…Extremal graph theory identifies what classes of graphs, or graph qualities, ensure certain minors are contained by a graph [32][33][34][35][36]. We will show that the final minor in the MSC of a complete bipartite graph is always the K N +1 graph, and in establishing the robustness of this minor for more general bipartite graphs, we turn to research on the development of theorems for the existence of complete minors: [37][38][39][40][41]. However the class of bipartite graphs under consideration are not particularly sparse, nor are they random, of large order, size, girth or degree.…”

Section: Introductionmentioning

confidence: 99%

Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets

Hamilton

Humble

2017

Quantum Inf Process

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Using quantum annealing to solve an optimization problem requires minor embeddings of a logic graph into a known hardware graph. In an effort to reduce the complexity of the minor embedding problem, we introduce the minor set cover (MSC) of a known graph G: a subset of graph minors which contain any remaining minor of the graph as a subgraph. Any graph that can be embedded into G will be embeddable into a member of the MSC. Focusing on embedding into the hardware graph of commercially available quantum annealers, we establish the MSC for a particular known virtual hardware, which is a complete bipartite graph. We show that the complete bipartite graph KN,N has a MSC of N minors, from which KN+1 is identified as the largest clique minor of KN,N . The case of determining the largest clique minor of hardware with faults is briefly discussed but remains an open question.

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Dense Minors In Graphs Of Large Girth

Cited by 21 publications

References 11 publications

Minors in Expanding Graphs

Minors in Expanding Graphs

Minors in graphs of large girth

Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets

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