The Exact Minimum Number of Triangles in Graphs With Given Order and Size

Liu, Hong; Pikhurko, Oleg; Staden, Katherine

doi:10.1017/fmp.2020.7

Cited by 17 publications

(14 citation statements)

References 31 publications

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“…Erdős [3] then extended this result to graphs with a linear number of extra edges and, in [4], studied the problem for larger cliques. Over the last fifty years, many further results in this direction were obtained by various researchers, see, e.g., [1,6,7,11,12,13] and their references. looks at graphs with at least ⌊n 2 /4⌋+ 1 edges, thus guaranteeing that there are always some triangles.…”

Section: Introductionmentioning

confidence: 93%

Books versus Triangles at the Extremal Density

Conlon¹,

Fox²,

Sudakov³

2020

SIAM J. Discrete Math.

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A celebrated result of Mantel shows that every graph on n vertices with ⌊n 2 /4⌋ + 1 edges must contain a triangle. A robust version of this result, due to Rademacher, says that there must in fact be at least ⌊n/2⌋ triangles in any such graph. Another strengthening, due to the combined efforts of many authors starting with Erdős, says that any such graph must have an edge which is contained in at least n/6 triangles. Following Mubayi, we study the interplay between these two results, that is, between the number of triangles in such graphs and their book number, the largest number of triangles sharing an edge. Among other results, Mubayi showed that for any 1/6 ≤ β < 1/4 there is γ > 0 such that any graph on n vertices with at least ⌊n 2 /4⌋ + 1 edges and book number at most βn contains at least (γ − o(1))n 3 triangles. He also asked for a more precise estimate for γ in terms of β. We make a conjecture about this dependency and prove this conjecture for β = 1/6 and for 0.24995 ≤ β < 1/4, thereby answering Mubayi's question in these ranges.The second question, about finding an edge in many triangles, was first studied by Erdős [3] in 1962. A book in a graph is a collection of triangles that have an edge in common. The size of the book is the number of such triangles. The book number of a graph G, denoted b(G), is the size of the largest book in the graph. Erdős proved that every graph G on n vertices with ⌊n 2 /4⌋ + 1 edges satisfies b(G) ≥ n/6 − O(1) and conjectured that the O(1)-term can be removed. Solving this conjecture and answering the second question above, Edwards and, independently, Khadžiivanov and Nikiforov [8] proved that every such graph satisfies b(G) ≥ n/6, which is tight.Our concern here is with a problem of Mubayi [10] about the interplay between the two questions above. More precisely, if a graph G on n vertices with ⌊n 2 /4⌋ + 1 edges satisfies b(G) ≤ b, at least how many triangles must it have? We write t(n, b) for this minimum number. Mubayi proved that for fixed β ∈ (1/4, 1/2), if b(G) < βn, then t(n, b) ≥ 1 2 β(1 − 2β) − o(1) n 2 , a bound which is asymptotically tight. He also showed that t(n, b) changes from quadratic to cubic in n when b ≈ n/4. More precisely, he proved that for each β ∈ (1/6, 1/4) there is γ > 0 such that t(n, βn) ≥ γn 3 . He then asked for a more precise determination of the optimal γ in terms of β, but added that the problem 'seems very hard'. Our contribution in this paper is to make a conjecture about this dependency and to confirm this conjecture for β = 1/6 and for 0.24995 ≤ β < 1/4.To say more, consider the 3-prism graph, the skeleton of the 3-prism, consisting of two disjoint triangles with a perfect matching between them. For nonnegative integers b and n with b ≤ n/4, let S b,n be the graph on n vertices formed by blowing up the 3-prism graph, where four of the six parts, corresponding to the vertices of two edges of the matching, are of size b, and the remaining two parts are of size ⌊(n − 4b)/2⌋ and ⌈(n − 4b)/2⌉. Restated, S b,n has vertex set consisting of ...

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Section: Introductionmentioning

confidence: 93%

Books versus Triangles at the Extremal Density

Conlon¹,

Fox²,

Sudakov³

2020

SIAM J. Discrete Math.

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“…A question reminiscent of the seminal result of Razborov on the minimal triangle density in graphs [25] (see also [22] and [20]) would be to determine the exact lower bound on ν d (G) in terms of d (answering asymptotically the question of Erdős stated above).…”

Section: Related Resultsmentioning

confidence: 99%

Sharp bounds for decomposing graphs into edges and triangles

Blumenthal

Lidický

Pehova

et al. 2020

Combinator. Probab. Comp.

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For a real constant α, let $\pi _3^\alpha (G)$ be the minimum of twice the number of K2’s plus α times the number of K3’s over all edge decompositions of G into copies of K2 and K3, where Kr denotes the complete graph on r vertices. Let $\pi _3^\alpha (n)$ be the maximum of $\pi _3^\alpha (G)$ over all graphs G with n vertices. The extremal function $\pi _3^3(n)$ was first studied by Győri and Tuza (Studia Sci. Math. Hungar.22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova (Combin. Probab. Comput.28 (2019) 465–472) proved via flag algebras that $\pi _3^3(n) \le (1/2 + o(1)){n^2}$ . We extend their result by determining the exact value of $\pi _3^\alpha (n)$ and the set of extremal graphs for all α and sufficiently large n. In particular, we show for α = 3 that Kn and the complete bipartite graph ${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$ are the only possible extremal examples for large n.

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“…Of course, minimising the number of r-cliques over (n, m)-graphs from the restricted class H is easier than the unrestricted version. The computation of H 3 (n, m) for all (n, m) appears in [11,Proposition 1.5]. Some large ranges of parameters when the conjecture has been proved are when m is slightly above t r (n) by Lovász and Simonovits [14] and when r = 3 and m/ n 2 is bounded away from 1 by Liu, Pikhurko and Staden [11].…”

Section: Introductionmentioning

confidence: 99%

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Kim¹,

Liu²,

Pikhurko³

et al. 2020

discrete_Anal

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The famous Erdős-Rademacher problem asks for the smallest number of r-cliques in a graph with the given number of vertices and edges. Despite decades of active attempts, the asymptotic value of this extremal function for all r was determined only recently, by Reiher [Annals of Mathematics, 184 (2016) 683-707]. Here we describe the asymptotic structure of all almost extremal graphs. This task for r = 3 was previously accomplished by Pikhurko and Razborov [Combinatorics, Probability and Computing, 26 (2017) 138-160].

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The Exact Minimum Number of Triangles in Graphs With Given Order and Size

Cited by 17 publications

References 31 publications

Books versus Triangles at the Extremal Density

Books versus Triangles at the Extremal Density

Sharp bounds for decomposing graphs into edges and triangles

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