Geodesic
A Note on R-Equitable K-Colorings of Trees
Yugoslav journal of operations research, Tome 24 (2014) no. 2, p. 293

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A graph $G = (V,E)$ is $r$-equitably $k$-colorable if there exists a partition of $V$ into $k$ independent sets $V_1, V_2,...,V_k$ such that $| |Vi| - |V_j| | \leq r$ for all $i, j \in \{1, 2, ..., k \}$. In this note, we show that if two trees $T_1$ and $T_2$ of order at least two are $r$-equitably $k$-colorable for $r \geq 1$ and $k \geq 3$, then all trees obtained by adding an arbitrary edge between $T_1$ and $T_2$ are also $r$-equitably $k$-colorable.
Classification : 05C15, 05C69.
Keywords: Trees, equitable coloring, independent sets. 
Alain Hertz; Bernard Ries. A Note on R-Equitable K-Colorings of Trees. Yugoslav journal of operations research, Tome 24 (2014) no. 2, p. 293 . http://geodesic.mathdoc.fr/item/YJOR_2014_24_2_a8/
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     title = {A {Note} on {R-Equitable} {K-Colorings} of {Trees}},
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     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/YJOR_2014_24_2_a8/}
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