On the number of unlabeled vertices in edge-friendly labelings of graphs

Let GG be a graph with vertex set V(G)V(G) and edge set E(G)E(G), and ff be a 0−10−1 labeling of E(G)E(G) so that the absolute difference in the number of edges labeled 11 and 00 is no more than one. Call such a labeling ffedge-friendly. We say an edge-friendly labeling induces a partial vertex labeling   if vertices which are incident to more edges labeled 11 than 00, are labeled 11, and vertices which are incident to more edges labeled 00 than 11, are labeled 00. Vertices that are incident to an equal number of edges of both labels we call unlabeled. Call a procedure on a labeled graph a label switching algorithm   if it consists of pairwise switches of labels. Given an edge-friendly labeling of KnKn, we show a label switching algorithm producing an edge-friendly relabeling of KnKn such that all the vertices are labeled. We call such a labeling opinionated.