Diracs theorem hamiltonian circuit diagram
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Hamiltonian path
Path in a graph that visits each vertex exactly once
This article is about the nature of Hamiltonian paths. For the question of the existence of a Hamiltonian path or cycle in a given graph, see Hamiltonian path problem.
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. The computational problems of determining whether such paths and cycles exist in graphs are NP-complete; see Hamiltonian path problem for details.
Hamiltonian paths and cycles are named after William Rowan Hamilton, who invented the icosian game, now also known as Hamilton
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Dirac's theorem on Hamiltonian Graphs
Overview & Background
Continuing the theme of gods post, we will look at a classic result in graph theory, namely Dirac’s theorem on Hamiltonian cycles. This result was first published by Gabriel A. Diracin 1952 , with later refinements bygd Ore, as well as Bondy and Chvátal. Three proofs of the theorem will be presented: the original proof from Dirac’s paper, an induction proof, and a short and direct proof. We will also consider the tightness of the bound as well as a generalization by Ore.
As a little historical sidenote before diving into the mathematics, Gabriel Dirac was the son of Margit Wigner, the sister of famous physicist Eugene Wigner. After a first marriage, from which Gabriel was born, Margit later remarried Eugene Wigner’s friend Paul Dirac, another equally famous physicist and one of the founders of Quantum Mechanics. Subsequently, Gabriel chose to take on the name Dirac. More interesting biographical details can be foun
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A NEW ALGORITHM FOR FINDINGHAMILTONIAN CIRCUITSASHAY DHARWADKER DISTINGUISHED PROFESSOR OF ENDOWED CHAIR INSTITUTE OF MATHEMATICS ashay@dharwadker.org |
We present a new polynomial-time algorithm for finding Hamiltonian circuits in graphs. It is shown that the algorithm always finds a Hamiltonian circuit in graphs that have at least three vertices and minimum degree at least half the total number of vertices. In the process, we also obtain a constructive proof of Dirac's famous theorem of 1952, for the first time. The algorithm finds a Hamiltonian circuit (respectively, tour) in all known examples of graphs that have a Hamiltonian circuit (respectively, tour). In view of the importance of the P versus NP question, we ask: does there exist a graph that has a Hamiltonian circuit (respectively, tour) b |