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In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points ) which are connected by edges (also called arcs , links or lines ).
The "pearls" of the title include theorems, proofs, problems, and examples in graph theory.The book has ten chapters; after an introductory chapter on basic definitions, the remaining chapters material on graph coloring; Hamiltonian cycles and Euler tours; extremal graph theory; subgraph counting problems including connections to permutations, derangements, and Cayley's formula; graph ...
Its authors have divided Elementary Number Theory, Group Theory and Ramanujan Graphs into four chapters. The first of these provides background in graph theory, including material on the girth of graphs (the length of the shortest cycle), on graph coloring, and on the use of the probabilistic method to prove the existence of graphs for which both the girth and the number of colors needed are ...
Biggs organised the book into four major sections; The Language of Mathematics, Techniques, Algorithms and Graphs, and Algebraic Methods. This book was an accumulation of Discrete Mathematics , first edition, textbook published in 1985 which dealt with calculations involving a finite number of steps rather than limiting processes.
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called vertices (also called nodes or points ) and each of the related pairs of vertices is called an edge (also called link or line ...
The De Bruijn–Erdős theorem for countable graphs can also be shown to be equivalent in axiomatic power, within a certain theory of second-order arithmetic, to Weak Kőnig's lemma. [ 16 ] For a counterexample to the theorem in models of set theory without choice, let G {\displaystyle G} be an infinite graph in which the vertices represent all ...
The telecommunications industry has also motivated advances in discrete mathematics, particularly in graph theory and information theory. Formal verification of statements in logic has been necessary for software development of safety-critical systems , and advances in automated theorem proving have been driven by this need.
The discharging method is used to prove that every graph in a certain class contains some subgraph from a specified list. The presence of the desired subgraph is then often used to prove a coloring result. [1] Most commonly, discharging is applied to planar graphs. Initially, a charge is assigned to each face and each vertex of the graph. The ...
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