Signed graphs and signed cycles of hyperoctahedral groups - Ryo Uchiumi (Electronic Journal of Graph Theory and Applications (EJGTA), 2023)
For a graph with edge ordering, a linear order on the edge set, we obtain a permutation of vertices by considering the edges as transpositions of endvertices. It is known from Dénes’ results that the permutation of a tree is a full cyclic for any edge ordering. As a corollary, Dénes counted up the number of representations of a full cyclic permutation by means of product of the minimal number of transpositions. Moreover, a graph with an edge ordering which the permutation is a full cyclic is characterized by graph embedding. In this article, we consider an analogy of these results for signed graphs and hyperoctahedral groups. We give a necessary and sufficient condition for a signed graph to have an edge ordering such that the permutation is an even (or odd) full cyclic. We show that the edge ordering of the signed tree with some loops always gives an even (or odd) full cyclic permutation and count up the number of representations of an odd full cyclic permutation by means of product of the minimal number of transpositions.