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Theorem 2.1 a connected graph has an euler tour if and only if every vertex has even degree A graph h is the line graph of some simple graph if and only if one can decompose the edges of h into cliques such that every vertex is in at most two cliques. Note that there are two things to prove

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That if the graph has an euler tour, then every vertex has even degree Ontains no other vertices beside these And if every vertex has even degree, then the graph has an euler tour.

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It covers simple graphs, multigraphs as well as their directed analogues, and more restrictive classes such as tournaments, trees and arborescences.

The only prerequisites to reading it are a basic knowledge of elementary set theory and matrix theory, although a further knowledge of abstract algebra and topology is needed for a few of the more difficult exercises The contents of this book may be conveniently divided into four parts. Since the edges in graphs with directed edges are ordered pairs, the definition of the degree of a vertex can be defined to reflect the number of edges with this vertex as the initial vertex and as the terminal vertex. The document discusses the topic of graph theory

It defines some key terms used in graph theory, such as vertices, edges, adjacency, and types of graphs. We will spend much of this first introduction to graph theory defining the terminology In graph theory, the term graph refers to a set of vertices and a set of edges A vertex can be used to represent any object

Graphs may contain undirected or directed edges.

In this section, we give the definitions of graphs, graphs’ properties, and the data structures that serve to contain information on the graph nodes and topology and that are used by almost all graph analysis algorithms. Chapter 10 introduction to graph theory loosely speaking, a graph is a collection of points called vertices and connecting segments called edges, each of which starts at a vertex, ends at a vertex and