On constructing 1-factors of labelled complete graph

Abstract Let p be a prime. Ball, Ortega-Moreno, and Prodromou asked about constructions of a 1-factor of a labelled $$K^{p+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>K</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> graph that is connected to a Hadamard-based 1-factorization of complete graphs. In this paper, we give some constructions on this problem, based on the primitive roots of $$\mathbb {F}_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> . In particular, we completely answered the problem for the case $$p\equiv 5\pmod {8}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>5</mml:mn> <mml:mspace/> <mml:mo>(</mml:mo> <mml:mo>mod</mml:mo> <mml:mspace/> <mml:mn>8</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> .