Growable realizations: a powerful approach to the Buratti-Horak-Rosa Conjecture - Ollis, M. A. (Ars mathematica contemporanea, 2022)
Label the vertices of the complete graph ?$K_v$? with the integers ?$\{0, 1, …, v-1\}$? and define the length of the edge between the vertices ?$x$? and ?$y$? to be ?$\min (|x-y|,v-|x-y|)$?. Let ?$L$? be a multiset of size ?$v-1$? with underlying set contained in ?$\{1, …, \lfloor v/2 \rfloor \}$?. The Buratti-Horak-Rosa Conjecture is that there is a Hamiltonian path in ?$K_v$? whose edge lengths are exactly ?$L$? if and only if for any divisor ?$d$? of ?$v$? the number of multiples of ?$d$? appearing in ?$L$? is at most ?$v-d$?. We introduce "growable realizations", which enable us to prove many new instances of the conjecture and to reprove known results in a simpler way. As examples of the new method, we give a complete solution when the underlying set is contained in ?$\{1, 4, 5\}$? or in ?$\{1, 2, 3, 4\}$? and a partial result when the underlying set has the form ?$\{1, x, 2x\}$?. We believe that for any set ?$U$? of positive integers there is a finite set of growable realizations that implies the truth of the Buratti-Horak-Rosa Conjecture for all but finitely many multisets with underlying set ?$U$?.
Keywords: complete graph; dolžina povezave; edge-length; growable realization; Hamiltonian path; Hamiltonova pot; polni graf; rastoča realizacija