/kloc-one of the famous classical mathematical problems in the 0/8th century. In a park in Konigsberg, there are seven bridges connecting two islands in the Fritz fritz pregl River and their banks (pictured). Is it possible to start from any of these four places, each bridge only passes once, and then return to the starting point? Euler studied and solved this problem in 1736. He simplified the problem to the "one stroke" problem shown on the right, which proved that the above method was impossible.
Hot issues in graph theory research. /kloc-At the beginning of the 8th century, the Fritz fritz pregl River crossed the town, and Naif Island was located in the river. There are seven bridges on the river, connecting the whole town. Local residents are keen on a difficult problem: is there a route that can cross seven bridges without repetition? This is the problem of the Seven Bridges in Konigsberg. L. Euler used points to represent islands and land, between which. Simplify rivers, islands and bridges into a network, and simplify the problem of seven bridges into the problem of judging whether the connected networks can draw a sum. He not only solved this problem, but also gave the necessary and sufficient conditions for a connected network to draw a stroke. If they are connected, the odd vertices (the number of arcs passing through this point is odd) are 0 or 2.
When Euler visited Konigsberg, Prussia (now Kaliningrad, Russia) in 1736, he found that local citizens were engaged in a very interesting pastime. In konigsberg, a river named Pregel runs through it. This interesting pastime is to walk across all seven bridges on Saturday. Each bridge can only cross once, and the starting point and ending point must be the same place.
Euler regarded every land as a point, and the bridge connecting the two lands was represented by a line.
Later, this way of walking could not be inferred. His argument is that every time a person enters a piece of land (or point) from one bridge, he (or she) leaves the point from another bridge in addition to the starting point. So every time he passes a point, there are two bridges (or lines) counted, and the line that leaves from the starting point and the line that finally returns to the starting point are also counted as two bridges, so the places are harmonious.
The graph formed by the seven bridges does not contain even numbers, so the above tasks cannot be completed.
Euler's consideration is very important and ingenious, which embodies the uniqueness of mathematicians in dealing with practical problems-abstracting a practical problem into a suitable "mathematical model". This research method is called "mathematical model method". There is no need to use any profound theory, but thinking is the key to solving difficult problems.
Next, based on a theorem in the network, Euler quickly judged that it was impossible to visit the seven bridges in Konigsberg at one time without repeating them. That is to say, for many years, the non-repetitive route that people have painstakingly searched for simply does not exist. A question that stumped so many people is such an unexpected answer!
1736, Euler expounded his method of solving problems in the paper report of "Seven Bridges in Konigsberg" submitted to Petersburg Academy of Sciences. His ingenious solution laid the foundation for the establishment of a new branch of mathematics-topology.
The Seven Bridges Problem and euler theorem Euler not only successfully answered the questions raised by Konigsberg residents, but also drew and proved three more extensive conclusions about a stroke, which people usually call euler theorem. For a connected graph, the path taken by a stroke from a node is usually called Euler path. People usually refer to the Euler path of a stroke returning to the starting point as Euler path. A graph with Euler paths is called an Euler graph.
This topic was included in the "Primary Mathematics" published by People's Education Publishing House, Volume 12. On page 95.
This topic has also been included in the first volume of junior high school by People's Education Edition. On page 12 1.
One stroke: ■⒈Any connected graph composed of even points can be drawn with one stroke. You can draw with any even point as the starting point, and finally you can finish drawing with this point as the end point.
■ 6. Any connected graph with only two singularities (the rest are even points) can be drawn with one stroke. When drawing, one singularity must be the starting point and the other singularity must be the end point.
■ [13] None of the other paintings can be drawn in one stroke. Divide the odd number by two to work out how many strokes are needed for this picture. )