Equitable Colorings of Vertex-Weighted Graphs
Siddharth Barman, Vignesh Viswanathan

TL;DR
This paper extends the Hajnal-Szemerédi theorem to vertex-weighted graphs, establishing new bounds and polynomial-time algorithms for equitable colorings with fairness guarantees.
Contribution
It introduces a generalized equitable coloring framework for weighted graphs, providing bounds, algorithms, and conditions for fairness in coloring and division problems.
Findings
Existence of $(1 + ext{epsilon})$-EQ1 colorings for large enough $k$
Existence of 2-EQ1 colorings for $k \,\geq\, \Delta + 1$
Polynomial-time algorithms for constructing such colorings
Abstract
We study a generalization of the classical Hajnal-Szemer\'edi theorem to vertex-weighted graphs. Given a graph with nonnegative vertex weights, a coloring is called -approximately equitable up to one vertex (-EQ1) if, for each color class, the total weight remaining after removing its maximum-weight vertex is at most times the weight of any other color class. For vertex-weighted graphs with maximum degree , we show that there exist instances for which no -coloring is -EQ1 for any and . In light of this impossibility, we relax these parameters and establish the following results for any vertex-weighted graph with maximum degree : (1) for any and all , there exists a -EQ1…
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