Improved Bounds for the Extremal Number of Subdivisions

Abstract

For a multigraph F , the k-subdivision of F is the graph obtained by replacing the edges of F with pairwise internally vertex-disjoint paths of length k + 1. Conlon and Lee conjectured that if k is even, then the (k − 1)-subdivision of any multigraph has extremal number O(n 1+ 1 k ), and moreover, that for any simple graph F there exists ε > 0 such that the (k − 1)-subdivision of F has extremal number O(n 1+ 1 k −ε ). In this paper, we prove both conjectures.

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.…”

“…By definition, if H is a subgraph of a subdivision, then it is bipartite, has no K 2,2 and all the vertices in one of its parts have degree at most two. The conjecture of Erdős was recently confirmed by Conlon and Lee [5], and in a stronger form by Janzer [13]. Conlon and Lee [5] further proposed the following more general conjecture.…”

“…Despite recent progress on this topic (see, e.g., [5,13,4,11,14,15,16]), this problem remains open for t ≥ 3. Moreover the following weaker form of the above conjecture, proposed by Conlon, Janzer and Lee [4] was open as well.…”

“…In this section, we present the proof of Theorem 4. Our proof works for all t ≥ 2 but since the case t = 2 is already known by [5,13], for computational convenience we assume that t ≥ 3.…”

The Extremal Number of the Subdivisions of the Complete Bipartite Graph

“…The estimation of the extremal number of subdivisions has attracted the attention of many researchers recently. See [2,3,[7][8][9][10][11][12][13][14] for results on the subject.…”