On universality of graphs with uniformly distributed edges
Discrete Mathematics
Random Ramsey graphs for the four-cycle
Discrete Mathematics
Tura´n's extremal problem in random graphs: forbidding even cycles
Journal of Combinatorial Theory Series B
Szemerédi's regularity lemma for sparse graphs
FoCM '97 Selected papers of a conference on Foundations of computational mathematics
Threshold functions for asymmetric Ramsey properties involving cycles
Random Structures & Algorithms
On triangle-free random graphs
Random Structures & Algorithms
The Regularity Lemma and Its Applications in Graph Theory
Theoretical Aspects of Computer Science, Advanced Lectures [First Summer School on Theoretical Aspects of Computer Science, Tehran, Iran, July 2000]
Regular pairs in sparse random graphs I
Random Structures & Algorithms
Turán's theorem in sparse random graphs
Random Structures & Algorithms
The Turán Theorem for Random Graphs
Combinatorics, Probability and Computing
K5-free subgraphs of random graphs
Random Structures & Algorithms
A probabilistic counting lemma for complete graphs
Random Structures & Algorithms
| Hi-index | 0.00 |
In this paper we investigate the behaviour of subgraphs of sparse @e-regular bipartite graphs G=(V"1@?V"2,E) with vanishing density d that are induced by small subsets of vertices. In particular, we show that, with overwhelming probability, a random set S@?V"1 of size s@?1/d contains a subset S^' with |S^'|=(1-@e^')|S| that induces together with V"2 an @e^'-regular bipartite graph of density (1+/-@e^')d, where @e^'-0 as @e-0. The necessity of passing to a subset S^' is demonstrated by a simple example. We give two applications of our methods and results. First, we show that, under a reasonable technical condition, ''robustly high-chromatic'' graphs contain small witnesses for their high chromatic number. Secondly, we give a structural result for almost all C"@?-free graphs on n vertices and m edges for odd @?, as long as m is not too small, and give some bounds on the number of such graphs for arbitrary @?.