MATHEMATICS OF COMPUTATION Volume 77, Number 263, July 2008, Pages 1831–1839 S 0025-5718(07)02063-7 Article electronically published on December 28, 2007 PARITY-REGULAR STEINHAUS GRAPHS MAXIME AUGIER AND SHALOM ELIAHOU Abstract. Steinhaus graphs on n vertices are certain simple graphs in bijec- tive correspondence with binary {0,1}-sequences of length n − 1. A conjecture of Dymacek in 1979 states that the only nontrivial regular Steinhaus graphs are those corresponding to the periodic binary sequences 110...110 of any length n − 1=3m. By an exhaustive search the conjecture was known to hold up to 25 vertices. We report here that it remains true up to 117 vertices. This is achieved by considering the weaker notion of parity-regular Steinhaus graphs, where all vertex degrees have the same parity. We show that these graphs can be parametrized by an F 2 -vector space of dimension approximately n/3 and thus constitute an efficiently describable domain where true regular Steinhaus graphs can be searched by computer. 1. Introduction Let s = a 1 ...a n−1 be a binary sequence of length n − 1 with entries a i in the 2-element field F 2 = {0, 1}. The Steinhaus graph associated with s is the simple graph G(s) on the vertex set {0, 1,...,n − 1} whose adjacency matrix M (s)= (a i,j ) ∈M n (F 2 ), with indices 0 ≤ i, j ≤ n − 1, is defined as follows: 1. a i,i =0 for 0 ≤ i ≤ n − 1, 2. a 0,i = a i for 1 ≤ i ≤ n − 1, 3. a i,j = a i−1,j−1 + a i−1,j for 1 ≤ i<j ≤ n − 1, 4. a j,i = a i,j for 0 ≤ i<j ≤ n − 1. Note that the first row of M (s) is the vector (0,a 1 ,...,a n−1 ), and that each subsequent row is determined, in its strict upper triangular part, by its predecessor using rule 3. For example, if s = a 1 ...a 4 , then M (s)= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0 a 1 a 2 a 3 a 4 a 1 0 a 1 + a 2 a 2 + a 3 a 3 + a 4 a 2 a 1 + a 2 0 a 1 + a 3 a 2 + a 4 a 3 a 2 + a 3 a 1 + a 3 0 a 1 + a 2 + a 3 + a 4 a 4 a 3 + a 4 a 2 + a 4 a 1 + a 2 + a 3 + a 4 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . The strict upper triangular part of M (s) is known as the Steinhaus triangle as- sociated with s, first defined by Steinhaus in [7]. We say that the graph G(s) is generated by the binary sequence s. Steinhaus graphs (in fact, their comple- ments) were introduced by John Molluzzo in [6]. It can easily be shown [3] that all Steinhaus graphs are connected, except those generated by the constant sequences 0...0. Received by the editor February 2, 2006 and, in revised form, April 13, 2007. 2000 Mathematics Subject Classification. Primary 11B75, 05C07, 05C50. c 2007 American Mathematical Society Reverts to public domain 28 years from publication 1831 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use