Books versus Triangles at the Extremal Density

Abstract

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 ...

Short proofs of some extremal results III

“…More generally, we can define graphs G = G n , r ( s ) for which G n , r is the case s = 1 and such that as we increase s from 1, the joint number j r + 1 ( G ) decreases, but the number of copies of K r + 1 increases. It also generalizes the construction that is conjectured by the authors in 26 to be tight for r = 2 and may give an asymptotically tight bound for general r . The graph G has n vertices and a vertex partition into r + 1 parts V 0 , V 1 , … , V r , with V 0 of size s and all the other parts of size or .…”

“…Mubayi 55 showed that t ( n , j ) jumps from quadratic to cubic in n when j goes below the threshold n /4. In 26, we studied this problem in more detail and proposed a precise conjecture that states that if n /6 ≤ j ≤ n /4, then every graph on n vertices with at least ⌊ n 2 /4⌋ edges and no edge in more than j triangles which is not the Turán graph T n , 2 has at least j 2 ( n − 4 j ) triangles. Moreover, we conjectured that equality holds if and only if the graph is S j , n , a blow‐up of the 3‐prism graph, the graph on six vertices which consists of two disjoint triangles with a perfect matching between them, with the parts corresponding to one of the matching edges of size ⌊( n − 4 j )/2⌋ and ⌈( n − 4 j )/2⌉ and the other four parts of size j .…”

“…Here we consider the generalization of Mubayi's problem to larger cliques. This question was raised by Mubayi 55 and again recently by the authors 26. Let t ( n , j , r ) be the minimum number of copies of K r in every graph G with n vertices, more than t r − 1 ( n ) edges and j r ( G ) ≤ j .…”

“…Proof of Theorem We may suppose r ≥ 3 since the case r = 2 is handled in 26 and 55. Let (so ) and let be a constant satisfying the assertion of Lemma 3.6.…”