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

TL;DR

This paper investigates the structure of edge-friendly labelings in graphs, especially focusing on the number of unlabeled vertices, and introduces a label switching algorithm to achieve fully labeled configurations in complete graphs.

Contribution

It presents a novel label switching algorithm that transforms edge-friendly labelings into opinionated labelings with no unlabeled vertices in complete graphs.

Findings

The algorithm guarantees all vertices are labeled in complete graphs.

It characterizes the conditions for unlabeled vertices in edge-friendly labelings.

Provides a method to convert partial labelings into complete labelings.

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 $f$ \emph{edge-friendly}. We say an edge-friendly labeling induces a \emph{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 \emph{unlabeled}. Call a procedure on a labeled graph a \emph{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 \textit{opinionated}.