Spreading of information, ideas or diseases can be conveniently modelled in the context of complex networks. An analysis now reveals that the most efficient spreaders are not always necessarily the most connected agents in a network. Instead, the position of an agent relative to the hierarchical topological organization of the network might be as important as its connectivity.

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Identification of influential spreaders in complex networks

This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.

Algorithms in Combinatorial Geometry

Extremal Graph Theory

The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

Digraphs Theory Algorithms And Applications

Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

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Graph Coloring Problems

Old and new problems and results in combinatorial number theory

Assumption University of Windsor sponsored a symposium for mathematicians from Ontario, Michigan, and Indiana, The symposium gave occasion for an informal lecture in which I discussed various old and new questions on number theory, geometry and analysis. In the following list, I record these problems, with the addition of references and of a few further questions. A. NUMBER THEORY 1. It is known [35, Vol. 1, Section 581 that n(2x) < 257(x) for sufficiently large x. Is it true that (1) n(x + y) < B(X) + n(y) ?-Ungrir has verified the inequality for y 5 41. Hardy and Littlewood [29, p. 691 have proved that (2) P(X + Y)-n(x) < cy/log y. In the same paper, they discuss many interesting conjectures. They put h-0 sup [R (x + Y)-n(x)1 = p(y) , x-00 and they conjecture that p(y) > y/log y, and that perhaps n(y)-p(y)-oo as y-) 00. Hardy and Littlewood deduce (2) by Bruds method. A very difficult conjecture, weaker than (1) but much stronger than (2), is that corresponding to each s > 0 there exists a yE such that, for y > yE, dx + Y)-dy) < (1 + dy/log y-It has not yet been disproved that p(y) = 1 for all y. If p(y) > 1 for some y, then lim inf (h+l-pn) < 00. 2. About seventy years ago, Piltz [38] conjectured that, for each c > 0, pn+l-pn= O(n&). Cram& conjectured [7, p. 241 that pn+l-pn = O((log n)?. If lim sup (p,+l-p,)/(log n) \" = 1 , then, for each E > 0, infinitely many of the intervals [n, n + (1-s)(log n) \" ] contain no primes, but for n > ns, there is a prime between n and n + (1 + E)(log n) \". The Riemann hypothesis implies that prl+1-pn < n&+1/2 [35, Vol. 1, p. 3381. Thus the old conjecture that there is always a prime between two consecutive squares already goes beyond the Riemann hypothesis.

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Some unsolved problems.

http://homepages.math.uic.edu/~suk/erdos_hajnal.pdf

Ramsey-type theorems

Fundamentals about Partition Relations. Trees and Positive Ordinary Partition Relations. Negative Ordinary Partition Relations and the Discussion of the Finite Case. The Canonization Lemmas. Large Cardinals. Discussion of the Ordinary Partition Relation with Superscript 2. Discussion of the Ordinary Partition Relation with Superscript < 3. Some Applications of Combinatorial Methods. A Brief Survey of the Square Bracket Relation.

Combinatorial Set Theory: Partition Relations for Cardinals

Let denote the class of all graphsG which satisfyG→(G
1,G
2). As a way of measuring minimality for members of
 , we define thesize Ramsey number ř(G
1,G
2) by
 . We then investigate various questions concerned with the asymptotic behaviour ofř.

Paul Erdős

Papers

Old and new problems and results in combinatorial number theory

Some unsolved problems.

Ramsey-type theorems

Combinatorial Set Theory: Partition Relations for Cardinals

The size ramsey number