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Hidden paths in graphs: How to find unique Hamiltonian paths?
In the field of mathematical graph theory, a Hamiltonian path (or traceable path) refers to a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cyclic path that visits each vertex once. Therefore, the discussion surrounding Hamiltonian paths is not only a mystery to mathematics enthusiasts, but also an important topic in information science and computing theory, because the problem of determining the existence of such paths and cycles is an NP-complete problem, meaning That is, it cannot be resolved within a reasonable time.
Hamiltonian paths and cycles have attracted widespread attention due to their importance in practical applications, such as robot navigation, transportation problems and circuit design.
The name of the Hamiltonian path comes from William Rowan Hamilton, who invented the "icosian game" (now called the Hamiltonian puzzle) to find the Hamiltonian cycle that forms in the edge graph of the dodecahedron. question. Although Hamilton solved this problem using icosian calculus, this solution cannot be generalized to arbitrary graphs. In fact, many mathematicians had studied the characteristics of Hamiltonian cycles in polyhedra long before his research.
Any graph containing a Hamiltonian path is called a traceable graph. If there is a Hamiltonian path passing through each pair of points, the graph is called a Hamiltonian connected graph. However, a Hamiltonian cycle can only form a loop extending between adjacent vertices.
A complete graph (more than two vertices) is a graph that necessarily contains a Hamiltonian cycle. Every cycle graph is also Hamiltonian.
A graph with a Hamiltonian cycle generally refers to a Hamiltonian graph, and any Hamiltonian cycle can be converted into a Hamiltonian path by removing an edge. But not all doubly connected graphs are guaranteed to be Hamiltonian. Since the 18th century, related research on Hamiltonian paths has been common, and can even be traced back to the early days of Indian mathematics.
For example, in the knight diagram of the chessboard, the problem of knight patrol has been discussed in Indian mathematics as early as the 9th century. Over time, the concept was further developed in Europe, with for example Abraham de Moivre and Leonhard Euler both exploring knightly patrols problem.
The diversification of Hamiltonian cycles has led mathematicians to conduct more in-depth research on its properties, such as graph density, toughness and taboo subgraphs.
In current research, the Bondy–Chvátal theorem provides the best vertex degree characteristics relative to Hamiltonian graphs, which enables most Hamiltonian judgments to be made quickly. These theories are not limited to random judgments, but are also closely related to the structure and characteristics of various graphs, allowing us to more clearly understand what kind of connection methods can achieve the establishment of Hamiltonian paths or loops in graphs of different properties.
According to existing research, the decomposition of any edge of the Hamiltonian graph G may form a Hamiltonian cycle. A more attractive application in practice is the Hamiltonian cycle polynomial, which is the graph description required in the weighted guided graph of the Hamiltonian cycle. If this polynomial is not constant zero in a specific case, it can be inferred that this Pictured is Hamilton.
When the existence of a Hamiltonian cycle becomes a difficult problem to explore, mathematicians begin to think of more efficient algorithms to solve such problems. Although there have been many achievements in theory, how to find an effective Hamiltonian path in practice is still an unsolved mystery.
Whether in mathematics or other application fields, the discussion of Hamiltonian paths and their existence is still deepening. This is not only a mathematical challenge, but also an important topic that promotes the advancement of computer science and logical thinking. Can you find the hidden Hamiltonian path in these complex graphs?
