Parameterized Complexity of Equitable Coloring
Discrete mathematics & theoretical computer science, ICGT 2018, Tome 21 (2019) no. 1
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A graph on $n$ vertices is equitably $k$-colorable if it is $k$-colorable and every color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. Such a problem appears to be considerably harder than vertex coloring, being $\mathsf{NP\text{-}Complete}$ even for cographs and interval graphs. In this work, we prove that it is $\mathsf{W[1]\text{-}Hard}$ for block graphs and for disjoint union of split graphs when parameterized by the number of colors; and $\mathsf{W[1]\text{-}Hard}$ for $K_{1,4}$-free interval graphs when parameterized by treewidth, number of colors and maximum degree, generalizing a result by Fellows et al. (2014) through a much simpler reduction. Using a previous result due to Dominique de Werra (1985), we establish a dichotomy for the complexity of equitable coloring of chordal graphs based on the size of the largest induced star. Finally, we show that \textsc{equitable coloring} is $\mathsf{FPT}$ when parameterized by the treewidth of the complement graph.
@article{DMTCS_2019_21_1_a4,
author = {Gomes, Guilherme de C. M. and Lima, Carlos V. G. C. and Santos, Vin{\'\i}cius F. dos},
title = {Parameterized {Complexity} of {Equitable} {Coloring}},
journal = {Discrete mathematics & theoretical computer science},
publisher = {mathdoc},
volume = {21},
number = {1},
year = {2019},
doi = {10.23638/DMTCS-21-1-8},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-1-8/}
}
TY - JOUR AU - Gomes, Guilherme de C. M. AU - Lima, Carlos V. G. C. AU - Santos, Vinícius F. dos TI - Parameterized Complexity of Equitable Coloring JO - Discrete mathematics & theoretical computer science PY - 2019 VL - 21 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-1-8/ DO - 10.23638/DMTCS-21-1-8 LA - en ID - DMTCS_2019_21_1_a4 ER -
%0 Journal Article %A Gomes, Guilherme de C. M. %A Lima, Carlos V. G. C. %A Santos, Vinícius F. dos %T Parameterized Complexity of Equitable Coloring %J Discrete mathematics & theoretical computer science %D 2019 %V 21 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-1-8/ %R 10.23638/DMTCS-21-1-8 %G en %F DMTCS_2019_21_1_a4
Gomes, Guilherme de C. M.; Lima, Carlos V. G. C.; Santos, Vinícius F. dos. Parameterized Complexity of Equitable Coloring. Discrete mathematics & theoretical computer science, ICGT 2018, Tome 21 (2019) no. 1. doi: 10.23638/DMTCS-21-1-8
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