On the Extremal Number of Subdivisions

Abstract

Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing K ′ s,t for the subdivision of the bipartite graph K s,t , we show that ex(n, K ′ s,t ) = O(n 3/2− 1 2s ). This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for t sufficiently large in terms of s. Second, for any integers s, k ≥ 1, we show that ex(n, L) = Θ(n 1+ s sk+1 ) for a particular graph L depending on s and k, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erdős and Simonovits, which asserts that every rational number r ∈ (1, 2) is realisable in the sense that ex(n, H) = Θ(n r ) for some appropriate graph H, giving infinitely many new realisable exponents and implying that 1 + 1/k is a limit point of realisable exponents for all k ≥ 1. Writing H k for the k-subdivision of a graph H, this result also implies that for any bipartite graph H and any k, there exists δ > 0 such that ex(n, H k−1 ) = O(n 1+1/k−δ ), partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph H with maximum degree r on one side which does not contain C 4 as a subgraph satisfies ex(n, H) = o(n 2−1/r ). . s = r. On the other hand, a recent conjecture of Conlon and Lee [5] says that containing K r,r as a subgraph should be the only reason why Theorem 1.1 is tight up to the constant.Conjecture 1.2 (Conlon-Lee). Let H be a bipartite graph such that in one of the parts all the degrees are at most r and H does not contain K r,r as a subgraph. Then there exists some δ > 0 such that ex(n, H) = O(n 2−1/r−δ ).To say more, recall that the k-subdivision of a graph L is the graph obtained by replacing the edges of L by internally disjoint paths of length k + 1. We shall write L k for the ksubdivision of L and L ′ for the 1-subdivision. It is easy to see that any C 4 -free bipartite graph in which every vertex in one part has degree at most two is a subgraph of K ′ t for some positive integer t. Conlon and Lee [5] verified their conjecture in the r = 2 case by proving the following result.Theorem 1.3 (Conlon-Lee). For any integer t ≥ 3, ex(n, K ′ t ) = O(n 3/2−1/6 t ).Our first result gives some small progress towards Conjecture 1.2 when r > 2.Theorem 1.4. Let H be a bipartite graph such that in one of the parts all the degrees are at most r and H does not contain C 4 as a subgraph. Then ex(n, H) = o(n 2−1/r ).

Turán number of bipartite graphs with no 𝐾_{𝑡,𝑡}

“…We first pass to a bipartite subgraph with parts V and W , where V is of order n, and |W | is of order n 1−1/t . This is in contrast with few previous papers in the same topic [4,5,13] which work with a bipartite subgraph G ′ of G in which both parts have roughly the same size. By setting the parameters correctly, the advantage of our first step is that the average size of the common neighborhood in V of the (t − 1)-tuples of vertices from W is some large constant.…”

“…For a hypergraph H, the subdivision of H is the bipartite graph H ′ whose two vertex classes are V (H) and E(H), and v ∈ V (H) and e ∈ E(H) are joined by an edge if v ∈ e. Then Conjecture 3 is equivalent to asking whether ex(n, H ′ ) = o(n 2−1/t ) for a subdivision H ′ of a t-uniform hypergraph. In [4], this conjecture is proved in the special case H is a linear hypergraph (that is, any two edges of H intersect in at most one vertex), which corresponds to the case in which the bipartite graph H is K 2,2 -free. Also, it is mentioned in [5] and [4] that Conjecture 3 holds in case H is the subdivision of the complete t-uniform hypergraph with t + 1 vertices, or the subdivision of a t-partite t-uniform hypergraph.…”

“…In the proof of our main theorem, we use the following technical lemma, which can be found as Lemma 2.2 in [4]. Here, a graph G is K-almost regular if the maximum degree of G is at most K-times the minimum degree.…”

The Extremal Number of the Subdivisions of the Complete Bipartite Graph

“…Moreover, they conjectured that ex(n, K 2 s,t ) = O(n 3 2 − 1 2s ) holds, which is tight for sufficiently large t by a general result of Bukh and Conlon [1] (see Theorem 4.1 below). The conjecture was proved by Conlon, Janzer and Lee [2]. About longer subdivisions, they proved the following result.…”

“…The first few results on this topic concerned the 2-subdivision of the complete bipartite graph. Conlon and Lee [3] proved that if s ≤ t, then ex(n, K 2 s,t ) = O(n 3 2 − 1 12t ). (Here, and everywhere else in the paper, it is assumed that n → ∞ and other parameters are kept constant.…”