Spanning Eulerian Subgraphs in Generalized Prisms

Abstract

For a graph \(G\) with vertices labeled \(1,2,\ldots,n\) and a permutation \(\alpha\) in \(S_n\), the symmetric group on \(\{1,2,\ldots,n\}\), the \(\alpha\)-generalized prism over \(G\), \(\alpha(G)\), consists of two copies of \(G\), say \(G_x\) and \(G_y\), along with the edges \((x_i, y_{\alpha(i)})\), for \(1 \leq i \leq n\). In [10], the importance of building large graphs by using generalized prisms is indicated. A graph \(G\) is supereulerian if it has a spanning eulerian subgraph. In this note, we consider results of the form that if \(G\) has property \(P\), then for any \(\alpha \in S_{|V(G)|}\), \(\alpha(G)\) is supereulerian. As a result, we obtain a few properties of \(G\) which implies that for any \(\alpha \in S_{|V(G)|}\), \(\alpha(G)\) is supereulerian. Also, while the permutations are restricted, the related result is discussed.