摘要
[目的]利用最省刻度尺的已有研究成果研究极小优美图的构造方法.[方法]对任意正整数n≥2,在长度是n的无刻度直尺上最少刻多少个刻度,就能度量1-n的所有长度,这就是最省刻度的尺子问题.给定正整数n,存在m个整数组成的集合{a_(i)},满足0=a_(1)<a_(2)<…<a_(m)=n,使得任意整数s(0≤s≤n)均可表示成该集合中两个元素的差a_(j)-a_(i),则称{a_(i)}为n上的受限差基.根据极小优美图和受限差基的定义,将极小优美图问题等效为最省刻度尺问题进而得到极小优美图的构造方法.[结果]由n≥5时K n不是优美图和n≥1时图K 4+K n,n是优美图的结论,得到了边数是6至82的极小优美图顶点数的上下界;用构造方法给出了图K_(3)∨K 1,3,n-3 e,K_(3,n)∨K_(3-e)和K_(2,3,n)∨K_(3)-7e的优美标号,从而证明了这三类图都是优美图,并且当0≤n≤9时,K_(3)∨K_(1,3,,n)-3 e和K_(2,3,n)∨K_(3)-7e都是极小优美图,当0≤n≤8时,K_(3,n)∨K_(3-e)都是极小优美图,由此给出了29组最省刻度尺的刻度值.[结论]最省刻度尺可以为构造极小优美图提供新的研究思路.
[Objective]For any positive integer n≥2,it is possible to measure all lengths from 1 to n by carving at least a few scales on an ungraduated ruler of length n.This is the problem of the ruler with the least number of scales.Given a positive integer n,there exists a set of m integers{a_(i)},which satisfies 0=a_(1)<a_(2)<…<a_(m)=n,so that any integer s(0≤s≤n)can be expressed as the difference a_(j)-a_(i) between the two elements in the set.Therefore,{a_(i)}is called the restricted difference basis on n.The ruler with the least number of divisions,restricted difference basis,and representation of graceful graphs are three unresolved mathematical problems.[Methods]According to the definitions of minimal graceful graphs and restricted difference basis,the construction of“ruler with the least number of divisions”,“minimal graceful graph”,and“restricted difference basis”is the same mathematical problem.[Results]The conclusion is that K n is not a graceful graph when n≥5,and K 4+Kn is a graceful graph when n≥1.The upper and lower bounds on the number of vertices of minimal graceful graphs with edges ranging from 6 to 82 are obtained;The graceful labels of graphs K_(3)∨K_(1,3,n-3e),K_(3,n)∨K_(3-e)and K_(2,3,n)∨K_(3)-7e are given using construction methods,thus proving that these three types of graphs are all graceful graphs.Moreover,when 0≤n≤9,K_(3)∨K 1,3,n-3e and K_(2,3,n)∨K_(3)-7e are all extremely graceful graphs.When 0≤n≤8,K_(3,n)∨K_(3-e)are all extremely graceful graphs,and 29 sets of scale values for the most economical rulers are given.[Conclusions]As the length n of the ruler increases,a set of scale values for this ruler needs to be calculated.Minimum scale value will become very difficult.At present,there is no literature on using the method of constructing graceful graphs to obtain the most economical scale value design.This article proposes a new approach to solve the problems of the most economical scale and restricted difference basis by using the method of constructing minimal graceful graphs.
作者
唐保祥
任韩
TANG Baoxiang;REN Han(School of Mathematics and Statistics,Tianshui Normal University,Tianshui 741001,China;School of Mathematics Sciences,East China Normal University,Shanghai 200062,China)
出处
《厦门大学学报（自然科学版）》
CAS
CSCD
北大核心
2024年第2期339-344,共6页
Journal of Xiamen University：Natural Science
基金
国家自然科学基金(11171114)。
关键词
最省刻度尺
优美图
联图
极小优美图
优美标号
least scale number on ruler
graceful graph
join graphs
minimal graceful graph
graceful labeling
作者简介
通信作者:唐保祥,tbx0618@sina.com。
