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

AbstractLet G be a graph with vertex set V(G) and edge set E(G), and f be a 0−1 labeling of E(G) so that the absolute difference in the number of edges labeled 1 and 0 is no more than one. Call such a labeling f edge-friendly. We say an edge-friendly labeling induces a partial vertex labeling if vertices which are incident to more edges labeled 1 than 0, are labeled 1, and vertices which are incident to more edges labeled 0 than 1, are labeled 0. 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 Kn, we show a label switching algorithm producing an edge-friendly relabeling of Kn such that all the vertices are labeled. We call such a labeling opinionated