Triangular imbeddings into surfaces of a join of equicardinal independent sets following an Eulerian graph

Abstract

If G is a graph and m ≥ 1 an integer, G_(m) is the graph obtained from G by replacing each vertex v by m vertices (v, 1), (v, 2), ..., (v, m) and joining two vertices (v, i) and (w, j) iff v and w are joined in G. G_(m) is a particular case of a join in the sense of J. L. Jolivet [5]. Suppose that G has a triangular imbedding into a surface S (orientable or not); can we find a triangular imbedding of $\overline G = G_{(m)}$into a surface $\overline S$which has the same orientability characteristic as S? We give some sufficient conditions to have an affirmative answer to this question.