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Modern graph theory
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This book presents an account of newer topics, including Szemer'edi's Regularity Lemma and its use; Shelah's extension of the Hales-Jewett Theorem; the precise nature of the phase transition in a random graph process; the connection between electrical networks and random walks on graphs; and the Tutte polynomial and its cousins in knot theory.Abstract:
The time has now come when graph theory should be part of the education of every serious student of mathematics and computer science, both for its own sake and to enhance the appreciation of mathematics as a whole. This book is an in-depth account of graph theory, written with such a student in mind; it reflects the current state of the subject and emphasizes connections with other branches of pure mathematics. The volume grew out of the author's earlier book, Graph Theory -- An Introductory Course, but its length is well over twice that of its predecessor, allowing it to reveal many exciting new developments in the subject. Recognizing that graph theory is one of several courses competing for the attention of a student, the book contains extensive descriptive passages designed to convey the flavor of the subject and to arouse interest. In addition to a modern treatment of the classical areas of graph theory such as coloring, matching, extremal theory, and algebraic graph theory, the book presents a detailed account of newer topics, including Szemer\'edi's Regularity Lemma and its use, Shelah's extension of the Hales-Jewett Theorem, the precise nature of the phase transition in a random graph process, the connection between electrical networks and random walks on graphs, and the Tutte polynomial and its cousins in knot theory. In no other branch of mathematics is it as vital to tackle and solve challenging exercises in order to master the subject. To this end, the book contains an unusually large number of well thought-out exercises: over 600 in total. Although some are straightforward, most of them are substantial, and others will stretch even the most able reader.read more
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Journal ArticleDOI
Optimal Superpositioning of Flexible Molecule Ensembles
Vytautas Gapsys,Bert L. de Groot +1 more
TL;DR: These new (to the authors' knowledge) superpositioning methods combine the benefits of variance and distance between nearest-neighbor(s) minimization, providing a solution for the analysis of intrinsic motions of flexible molecules and resolving ambiguous rotations.
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Perfect matchings in 3-partite 3-uniform hypergraphs
Allan Lo,Klas Markström +1 more
TL;DR: A Dirac-type vertex degree threshold for perfect matchings in 3-partite 3-uniform hypergraphs is determined.
Journal ArticleDOI
H-coloring tori
John Engbers,David Galvin +1 more
TL;DR: In this paper, a structural characterization of the space of H-colorings of Z"m^d with fixed m and fixed n is given. But the analysis of the entropy of f is restricted to the case of n = 2 and H = n.
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Tutte's barycenter method applied to isotopies
TL;DR: This paper presents a method for building isotopies of triangulations in the plane, based on Tutte's theorem and the computation of equilibrium stresses of graphs by Maxwell-Cremona's theorem, and provides a counterexample showing that the analogue of Tute's theorem in dimension 3 is false.
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Stability of stationary interfaces of binary-tree type
Ryo Ikota,Eiji Yanagida +1 more
TL;DR: In this article, the curvature-driven motion of an interface on a bounded domain that contacts with the boundary at the right angle and has triple junctions with prescribed angles is considered.