Equitable Colorings of Vertex-Weighted Graphs

We study a generalization of the classical Hajnal-Szemerédi theorem to vertex-weighted graphs. Given a graph with nonnegative vertex weights, a coloring is called $\alpha$-approximately equitable up to one vertex ($\alpha$-EQ1) if, for each color class, the total weight remaining after removing its maximum-weight vertex is at most $\alpha \ge 1$ times the weight of any other color class. For vertex-weighted graphs with maximum degree $\mathrm{\Delta }$, we show that there exist instances for which no $k$-coloring is $\alpha$-EQ1 for any $k<\frac{3\mathrm{\Delta }}{2}$ and $\alpha <\sqrt{2}$. In light of this impossibility, we relax these parameters and establish the following results for any vertex-weighted graph $G$ with maximum degree $\mathrm{\Delta }$: (1) for any $\epsilon \in (0,1)$ and all $k\ge (\frac{c}{{\epsilon }^2}\ln \frac{1}{\epsilon })\mathrm{\Delta }$, there exists a $(1+\epsilon )$-EQ1 $k$-coloring of $G$, where $c$ is a fixed constant; and (2) for all $k\ge \mathrm{\Delta }+1$, there exists a $2$-EQ1 $k$-coloring of $G$. Furthermore, such equitable colorings can be computed in polynomial time. En route to our results on equitability under vertex weights, we establish sufficient conditions for the existence of $k$-colorings that are equitable with respect to any given partition of the vertex set. Our coloring results correspond to fairness guarantees in a constrained fair division setting and lead to concentration inequalities for partly dependent random variables.