Friendly-index set

In graph theory, a friendly-index set is a finite set of integers associated with a given undirected graph and generated by a type of graph labeling called a friendly labeling.

A friendly labeling of an $n$-vertex undirected graph $G = (V,E)$ is defined to be an assignment of the values 0 and 1 to the vertices of $G$ with the property that the number of vertices labeled 0 is as close as possible to the number of vertices labeled 1: they should either be equal (for graphs with an even number of vertices) or differ by one (for graphs with an odd number of vertices).

Given a friendly labeling of the vertices of $G$, one may also label the edges: a given edge $uv$ is labeled with a 0 if its endpoints $u$ and $v$ have equal labels, and it is labeled with a 1 if its endpoints have different labels. The friendly index of the labeling is the absolute value of the difference between the number of edges labeled 0 and the number of edges labeled 1.

The friendly index set of $G$, denoted $FI(G)$, is the set of numbers that can arise as friendly indexes of friendly labelings of $G$.

The Dynamic Survey of Graph Labeling contains a list of papers that examines the friendly indices of various graphs.