Equitable Colorings of Vertex-Weighted Graphs

2026-05-10

AI Summary

This study addresses the approximate fair coloring problem on vertex-weighted graphs, where fairness is defined by balancing the total remaining weight across color classes after removing the heaviest vertex from each class. The work introduces the α-EQ1 fairness criterion and extends the classical Hajnal–Szemerédi theorem to weighted graphs for the first time. By combining graph-theoretic analysis, probabilistic methods, combinatorial optimization, and concentration inequalities for partially dependent random variables, the authors prove that a 2-EQ1 coloring exists whenever the number of colors $k$ satisfies $k \geq \Delta + 1$, and that a $(1+\varepsilon)$-EQ1 coloring is achievable for any $\varepsilon \in (0,1)$ when $k$ is sufficiently large. All results are constructively obtained in polynomial time, establishing a theoretical bridge between existence and algorithmic feasibility for this fairness notion.

Abstract

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 $α$-approximately equitable up to one vertex ($α$-EQ1) if, for each color class, the total weight remaining after removing its maximum-weight vertex is at most $α\geq 1$ times the weight of any other color class. For vertex-weighted graphs with maximum degree $Δ$, we show that there exist instances for which no $k$-coloring is $α$-EQ1 for any $k < \frac{3Δ}{2}$ and $α< \sqrt{2}$. In light of this impossibility, we relax these parameters and establish the following results for any vertex-weighted graph $G$ with maximum degree $Δ$: (1) for any $\varepsilon \in (0,1)$ and all $k \geq (\frac{c}{\varepsilon^2}\ln{\frac{1}{\varepsilon}}) Δ$, there exists a $(1 + \varepsilon)$-EQ1 $k$-coloring of $G$, where $c$ is a fixed constant; and (2) for all $k \ge Δ+ 1$, 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.

Problem

• equitable coloring
• vertex-weighted graphs
• fair division
• graph coloring
• approximate fairness

Innovation

• equitable coloring
• vertex-weighted graphs
• approximate fairness
• polynomial-time algorithm
• Hajnal-Szemerédi theorem