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

Abstract

Let 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 K n , we show a label switching algorithm producing an edge-friendly relabeling of K n such that all the vertices are labeled. We call such a labeling opinionated.