On some interconnections between combinatorial optimization and extremal graph theory

The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions

Extensions of Extremal Graph Theory to Grids

We consider extensions of Turán\u27s original theorem of 1941 to planar grids. For a complete kxm array of vertices, we establish in Proposition 4.3 an exact formula for the maximal number of edges possible without any square regions. We establish with Theorem 4.12 an upper bound and with Theorem 4.15 an asymptotic lower bound for the maximal number of edges on a general grid graph with n vertices and no rectangles

Graphs with large maximum degree containing no odd cycles of a given length

Let us write f(n, Δ; C2k+1) for the maximal number of edges in a graph of order n and maximum degree Δ that contains no cycles of length 2k + 1. For n/2 ≤ Δ ≤ ?n - k - 1 and n sufficiently large we show that f(n,Δ; C2k+1) = Δ(n - Δ), with the unique extremal graph a complete bipartite graph. © 2002 Published by Elsevier Science (USA)

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