汕头大学学报（自然科学版）

任意正整数l≥2，在长度为l的无刻度直尺上最少刻多少个刻度可度量1至l的所有整数长度，这是至今未解决的最省刻度尺问题.现有研究依托高效算法与高性能计算资源，已获取长度1-244的最省刻度尺的部分最省刻度值，是当前数据连续最完整的成果，但该类方法耗时耗力，不仅仍遗漏大量最省刻度数据，且相关成果尚未形成系统理论.本文明确最省刻度尺设计与极小优美图构造的等价关系，填补二者研究关联的空白；用构造法给出图K_1,4,n∨K₄-9e、K_2,3,n∨K₄-3e、K_4,n∨K₄-4e、K_1,1,3,n∨K₃-9e和K_1,1,3,3,n-7e的优美标号，证明这五类图均为优美图，确定n取特定值时的部分极小优美图，由此得到长度24至92间37组最省刻度尺刻度数值，部分结果未见于现有文献.该方法无需依赖算力，理论上可拓展至更长长度的刻度求解，既为最省刻度尺问题提供新路径，也丰富了优美图的应用场景，兼具重要理论价值与实践指导意义.

For any positive integer l ≥2, the minimum ruler problem refers to determining the least number of marks required on an unmarked straightedge of length l to measure all integer lengths from 1 to l, which remains an unsolved mathematical problem to date. Existing studies,relying on high-efficiency algorithms and high-performance computing resources, have derived partial optimal mark values of minimum rulers for lengths ranging from 1 to 244, yielding the most complete continuous dataset available so far. Nevertheless, such methods are both timeconsuming and resource-intensive, leaving a large number of optimal mark values unobtained,and the relevant findings have not yet been integrated into a systematic theoretical framework.This paper clarifies the equivalence between the design of minimum rulers and the construction of minimal graceful graphs, thus bridging the gap in the research nexus between these two domains. By adopting a constructive approach, we assign graceful labelings to the graphs K_1,4,n∨K₄-9e、K_2,3,n∨K₄-3e、K_4,n∨K₄-4e、K_1,1,3,n∨K₃-9e and K_1,1,3,3,n-7e, proving that these five graph classes are all graceful graphs, and identifying some minimal graceful graphs within them when n takes specific values. On this basis, 37 sets of optimal mark values of minimum rulers for lengths from 24 to 92 is obtained, part of which have not been documented in existing literature.The proposed method is independent of computing power, and can theoretically be extended to calculate optimal mark values for longer ruler lengths. It not only offers a novel research avenue for the minimum ruler problem, but also enriches the application scenarios of graceful graphs,hence bearing important theoretical value and practical guiding significance.