graph theory

graph theory

Mathematical theory of networks. A graph consists of vertices (also called points or nodes) and edges (lines) connecting certain pairs of vertices. An edge that connects a node to itself is called a loop. In 1735 Leonhard Euler published an analysis of an old puzzle concerning the possibility of crossing every one of seven bridges (no bridge twice) that span a forked river flowing past an island. Euler's proof that no such path exists and his generalization of the problem to all possible networks are now recognized as the origin of both graph theory and topology. Since the mid-20th century, graph theory has become a standard tool for analyzing and designing communications networks, power transmission systems, transportation networks, and computer architectures.

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In graph theory, the girth of a graph is the length of a shortest cycle contained in the graph. If the graph does not contain any cycles, its girth is defined to be infinity. For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3. A graph with girth ≥ 4 is triangle-free.

Cages

A cubic graph of girth g that is as small as possible is known as a g-cage. The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, and the Tutte eight cage is the unique 8-cage.

Girth and graph coloring

For any positive integers g and χ, there exists a graph with girth at least g and chromatic number at least χ; for instance, the Grötzsch graph is triangle-free and has chromatic number 4, and repeating the Mycielskian construction used to form the Grötzsch graph produces triangle-free graphs of arbitrarily large chromatic number. Paul Erdős was the first to prove the general result, using the probabilistic method. More precisely, he showed that a random graph on n vertices, formed by choosing independently whether to include each edge with probability n(1 − g)/g, has, with probability tending to 1 as n goes to infinity, at most n/2 cycles of length g or less, but has no independent set of size n/2k. Therefore, removing one vertex from each short cycle leaves a smaller graph with girth greater than g, in which each color class of a coloring must be small and which therefore requires at least k colors in any coloring.

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References

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