On <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e41" altimg="si3.svg"><mml:mi>λ</mml:mi></mml:math>-fold relative Heffter arrays and biembedding multigraphs on surfaces

In this paper we define a new class of partially filled arrays, called λ-fold relative Heffter that are generalization the arrays introduced by Archdeacon in 2015. After showing connection concept with several other ones, such as signed magic graph decompositions and difference families, determine some necessary conditions present existence results for infinite classes these arrays. last part also show give rise to biembeddings multigraphs into orientable surfaces provide families biembeddings. To conclude, result concerning pairs covering surfaces.