Friendly-index set

File:Friendly index.svg

The number of vertices here is even, so a friendly labeling of the vertices means assigning an equal number of 1's (blue) and 0's (red) to the vertices of the graph. In a graph with an odd number of vertices, there would be exactly one more of either number than the other. Each edge is labeled 0 if the two vertices it connects are the same number, and 1 if they are different.

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 Template:Mvar-vertex undirected graph G = (V,E) is defined to be an assignment of the values 0 and 1 to the vertices of Template:Mvar 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 Template:Mvar, one may also label the edges: a given edge Template:Mvar is labeled with a 0 if its endpoints Template:Mvar and Template:Mvar 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 Template:Mvar, denoted FI(G), is the set of numbers that can arise as friendly indexes of friendly labelings of Template:Mvar.

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

References

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