Graph Theory

Posters on basic problems in Graph Theory Posters done by the students presenting some problems from Graph Theory, for example: The Travelling Salesman Problem, The Seven Bridges of Kรถnigsberg or Four Colour Theorem

What is graph theory?

The mathematical study of the properties of the formal mathematical structures called graphs.

What is a graph?

In a mathematician's terminology, a graph is a collection of points and lines connecting some subset of them. The points of a graph are most commonly known as graph vertices, but may also be called "nodes" or simply "points." Similarly, the lines connecting the vertices of a graph are most commonly known as graph edges, but may also be called "arcs" or "lines."

The study of graphs is known as graph theory, and was first systematically investigated by D. König in the 1930s. Euler's proof of the nonexistence of a so-called Eulerian cycle across all seven bridges of Königsberg, now known as the Königsberg bridge problem, is a famous precursor to graph theory. In fact, the study of various sorts of paths in graphs (e.g., Eulerian paths, Eulerian cycles, Hamiltonian paths, and Hamiltonian cycles) has many applications in real-world problems.

Topics: •

The Königsberg bridge problem The Königsberg bridge problem asks if the seven bridges of the city of Königsberg, formerly in Germany but now known as Kaliningrad and part of Russia, over the river Preger can all be traversed in a single trip without doubling back, with the additional requirement that the trip ends in the same place it began. This is equivalent to asking if the multigraph on four nodes and seven edges has an Eulerian cycle. This problem was answered in the negative by Euler (1736), and represented the beginning of graph theory.

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Chinese Postman Problem A problem asking for the shortest tour of a graph which visits each edge at least once. For an Eulerian graph, an Eulerian cycle is the optimal solution.

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Traveling Salesman Problem A problem requiring the most efficient (i.e., least total distance) Hamiltonian cycle a salesman can take through each of cities. No general method of solution is known.

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The four colour theorem The four-colour theorem states that any map in a plane can be coloured using four colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color.

Source: http://mathworld.wolfram.com/

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