Let G = ( g 1 , … , g ℓ ) be a vector of positive integers and set n = ∑ i = 1 ℓ g i . Writing Ch ( a , b ) for the Christoffel word with parameters ( a , b ) , we study the following synchronization problem: choose one conjugate of each word Ch ( g i , n − g i ) so that, at every position 0 , … , n − 1 , exactly one of the chosen words contains the letter 1. This gives a gap-free form of superimposition, motivated by Fraenkel’s conjecture. We encode the choice of conjugates by a shift vector V = ( v 1 , … , v ℓ ) . On the cyclic Cayley graph of Z / n Z , this yields an orbital matrix O ( G , V ) , with entries o i , j = ( v i + j g i ) mod n , and a Christoffel–conjugate matrix C ( G , V ) , which records the columns where a wraparound occurs. The vector V is a synchronizing seed precisely when each column of C ( G , V ) contains exactly one nonzero entry. The main algebraic tool is a vertical invariant: the column sums of O ( G , V ) are constant if and only if this one-wraparound-per-column condition holds. Using this criterion, we give explicit synchronizing seeds for every pair of Christoffel words of common length, for triples in which two generators coincide, for triples with three equal generators, and for the all-distinct family proportional to ( 1,2,4 ) . Finally, we give a geometric interpretation: each synchronized row is the Freeman chain code of a standard 4-connected Réveillès segment, whose parameter μ i is the unique integer in { 1 − n , … , 0 } satisfying μ i ≡ − v i ( mod n ) .

Abstract Given a 3-uniform hypergraph H having a set V of vertices, and a set of hyperedges $$T\subset \mathcal {P}(V)$$ T ⊂ P ( V ) , whose elements have cardinality three each, a null labelling is an assignment of $$\pm 1$$ ± 1 to the hyperedges such that each vertex belongs to the same number of hyperedges labelled $$+1$$ + 1 and $$-1$$ - 1 . A sufficient condition for the existence of a null labelling of H (proved in Di Marco et al. Lect Notes Comput Sci 12757:282–294, 2021) is a Hamiltonian cycle in its 2-intersection graph. The notion of 2-intersection graph generalizes that of intersection graph of an (hyper)graph and extends its effectiveness. The present study first shows that this sufficient condition for the existence of a null labelling in H can not be weakened by requiring only the connectedness of the 2-intersection graph. Then some interesting properties related to their clique configurations are proved. Finally, the main result is proved, the NP-completeness of this characterization and, as a consequence, of the construction of the related 3-hypergraphs.