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

Abstract

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 $\mathcal{A}(G)$ be the average number of colors in the non-equivalent colorings of a graph $G$. We give a general upper bound on $\mathcal{A}(G)$ that is valid for all graphs $G$ and a more precise one for graphs $G$ of order $n$ and maximum degree $\Delta (G)\in \{1,2,n-2\}$.

21 pages, 1 figure. arXiv admin note: text overlap with arXiv:2104.14172

References in corpus (2)

• Using Graph Theory to Derive Inequalities for the Bell Numbers
• Lower Bounds and properties for the average number of colors in the non-equivalent colorings of a graph