【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=a₁<a₂<…<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K₄+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K₃∨K_1,3,n-3e,K_3,n∨K₃-e和K_2,3,n∨K₃-7e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，K₃∨K_1,3,,n-3e和K_2,3,n∨K₃-7e都是极小优美图，当0≤n≤8时，K_3,n∨K₃-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.更多还原

【Abstract】 [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=a₁<a₂<…<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 K₄+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 K₃∨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,K₃∨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.更多还原