基于加权图的经典Ramsey数R(3,4)>8

Khairul Azmi, Elva, Widia
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引用次数: 0

摘要

目前,拉姆齐数的研究已经扩展到更广泛的范围,不仅局限于两个完全图之间的互补关系,还包括完全图、圆图、星图、轮图等的组合。虽然经典的拉姆齐数字仍然留下了需要解决的问题。拉姆齐数R(3,4) > 8。这意味着m=8是最大的整数,使得K_(8)包含一个红色图G和一个完整的蓝色图G的分量,仍然有可能在图G中不得到K_3,在图G中不得到蓝色K_4,图K_(8)总共有28条边。为了不得到红色的K_3边对,需要避免的红色边对的组合有(28,3)个。有(28,4)条边对的组合。为了不得到一对蓝色的K_4边,这需要避免。直接确定图的颜色当然是非常困难的,特别是当拉姆齐数越来越大的时候。这就像大海捞针一样。需要用特殊的方法才能解决这个问题。加权图法可以解决这一问题,该方法为每条边赋予一定的权值。加权图法可以将图K_8以阶为8×8的G矩阵的形式显示出来。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The Classical Ramsey Number R(3,4)>8 Using the Weighted Graph
At present, research on Ramsey Numbers has expanded to a wider scope, not only between 2 complete graphs that are complementary to each other but also a combination of complete graphs, circle graphs, star graphs, wheel graphs, and others. While the classic Ramsey number still leaves problems that need to be solved. Ramsey number R(3,4) > 8. This means that m=8 is the largest integer such that K_(8) which contains components of a red graph G and a complete blue graph G which is still possible not to get K_3 in graph G and not get a blue K_4 in graph G .The graph K_(8) has a total of 28 edges. There are as many as the combination (28,3) red edge pairs that need to be avoided so as not to get any red K_3 edge pairs. And there are as many as the combination (28,4) edge pairs  blue. That needs to be avoided in order not to get a pair of blue K_4 edges. Determining the coloring of the of the graph directly is certainly very difficult, especially if the Ramsey number is getting bigger. It's like looking for a needle in a haystack. Need to use a special method in order to solve this problem. The weighting graph method, where each edge is given weight with a certain value, is able to solve this problem. The weighting graph method is able to display the graph K_8 in the form of a G matrix with the order of 8×8.
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