On γ-labeling the almost-bipartite graph Pm+e

An almost-bipartite graph is a non-bipartite graph with the property that the removal of a particular single edge renders the graph bipartite. A graph labeling of an almost-bipartite graph G with n edges that yields cyclic G-decompositions of the complete graph K2nt+1 was recently introduced by Blinco, El-Zanati, and Vanden Eynden. They called such a labeling a γ-labeling. Here we show that the class of almost-bipartite graphs obtained from a path with at least 3 edges by adding an edge joining distinct vertices of the path an even distance apart has a γ-labeling.