Stability in the Erdős–Gallai Theorems on cycles and paths: Dedicated to the memory of G.N. Kopylov

The Erdős–Gallai Theorem states that for k≥2, every graph of average degree more than k−2 contains a k-vertex path. This result is a consequence of a stronger result of Kopylov: if k is odd, k=2t+1≥5, n≥(5t−3)/2, and G is an n-vertex 2-connected graph with at least h(n,k,t):=(k−t2)+t(n−k+t) edges, then G contains a cycle of length at least k unless G=Hn,k,t:=Kn−E(Kn−t). In this paper we prove a stability version of the Erdős–Gallai Theorem: we show that for all n≥3t>3, and k∈{2t+1,2t+2}, every n-vertex 2-connected graph G with e(G)>h(n,k,t−1) either contains a cycle of length at least k or contains a set of t vertices whose removal gives a star forest. In particular, if k=2t+1≠7, we show G⊆Hn,k,t. The lower bound e(G)>h(n,k,t−1) in these results is tight and is smaller than Kopylov's bound h(n,k,t) by a term of n−t−O(1). © 2016 Elsevier Inc

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

On Even-Degree Subgraphs of Linear Hypergraphs

A subgraph of a hypergraph H is even if all its degrees are positive even integers, and b-bounded if it has maximum degree at most b. Let f(b)(n) denote the maximum number of edges in a linear n-vertex 3-uniform hypergraph which does not contain a b-bounded even subgraph. In this paper, we show that if b >= 12, then n log n/3b log log n <= f(b)(n) <= Bn(log n)(2) for some absolute constant B, thus establishing f(b)(n) up to polylogarithmic factors. This leaves open the interesting case b = 2, which is the case of 2-regular subgraphs. We are able to show for some constants c, C > 0 that cn log n <= f2( n) <= Cn(3/2)(log n)(5). We conjecture that f(2)(n) = n(1+o(1)) as n -> infinity