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 $G$ .^[1]

The Dynamic Survey of Graph Labeling contains a list of papers that examines the friendly indices of various graphs.^[2]

References

^ Kwong, Harris; Lee, Sin-Min; Ng, Ho (2008). "On friendly index sets of 2-regular graphs". Discrete Math. 308 (23): 5522–5532. doi:10.1016/j.disc.2007.10.018. MR 2459372.
^ Gallian, Joseph A (2009). "A dynamic survey of graph labelling" (PDF). El. J. Combinat. 16 (#DS6).

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[1] Kwong, Harris; Lee, Sin-Min; Ng, Ho (2008). "On friendly index sets of 2-regular graphs". Discrete Math. 308 (23): 5522–5532. doi:10.1016/j.disc.2007.10.018. MR 2459372.

[2] Gallian, Joseph A (2009). "A dynamic survey of graph labelling" (PDF). El. J. Combinat. 16 (#DS6).

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