Group for Research in Decision Analysis

# Upper bounds on the average number of colors in the non-equivalent colorings of a graph

## Alain Hertz, Hadrien Mélot, Sébastien Bonte, Gauvain Devillez, and Pierre Hauweele

A coloring of a graph is an assignment of colors to its vertices such that adjacent vertices have different colors. Two colorings are equivalent if they induce the same partition of the vertex set into color classes. Let $$A(G)$$ be the average number of colors in the non-equivalent colorings of a graph $$G$$. We give a general upper bound on $$A(G)$$ that is valid for all graphs $$G$$ and a more precise one for graphs $$G$$ of order $$n$$ and maximum degree $$\Delta(G)\in \{1,2,n-2\}$$.

, 18 pages