Fine-Wilf graphs and the generalized Fine-Wilf theorem (Q2855629)

The Fine-Wilf theorem [\textit{N. J. Fine} and \textit{H. S. Wilf}, Proc. Am. Math. Soc. 16, 109--114 (1965; Zbl 0131.30203)] states that, if two periodic sequences of respective periods \((r)\) and \((s)\) coincide on their first \((r+s-\gcd(r,s))\) terms, they must be equal (and this number of terms is optimal). This result can be translated into a property of finite sequences with two periods, that was generalized to finite sequences with three periods by \textit{M. G. Castelli}, \textit{F. Mignosi} and \textit{A. Restivo} [Theor. Comput. Sci. 218, No. 1, 83--94 (1999; Zbl 0916.68114)] and infinite sequences with an arbitrary number of periods by \textit{J. Justin} [Theor. Inform. Appl. 34, No. 5, 373--377 (2000; Zbl 0987.68056)]. \textit{R. Tijdeman} and \textit{L. Q. Zamboni} [Indag. Math., New Ser. 14, No. 1, 135--147 (2003; Zbl 1091.68088); Theor. Comput. Sci. 410, No. 30--32, 3027--3034 (2009; Zbl 1173.68056) and Integers 9, No. 3, 333--342, A27 (2009; Zbl 1193.68204)] generalized the work of Castelli, Mignosi and Restivo [loc. cit.]. The authors of the present paper give a reformulation of the results of Castelli-Mignosi-Restivo [loc. cit.] and Tijdeman-Zamboni [loc. cit.] obtaining new interesting properties.