# Graph colorings with restricted bicolored subgraphs: I. Acyclic, star, and treewidth colorings

@article{Bradshaw2022GraphCW,
title={Graph colorings with restricted bicolored subgraphs: I. Acyclic, star, and treewidth colorings},
journal={Journal of Graph Theory},
year={2022},
volume={100},
pages={362 - 370}
}
• Published 30 August 2020
• Mathematics
• Journal of Graph Theory
We show that for any fixed integer m ≥ 1 $m\ge 1$ , a graph of maximum defiggree Δ ${\rm{\Delta }}$ has a coloring with O ( Δ ( m + 1 ) ∕ m ) $O({{\rm{\Delta }}}^{(m+1)\unicode{x02215}m})$ colors in which every connected bicolored subgraph contains at most m $m$ edges. This result unifies previously known upper bounds on the number of colors sufficient for certain types of graph colorings, including star colorings, for which O ( Δ 3 ∕ 2 ) $O({{\rm{\Delta }}}^{3\unicode{x02215}2})$ colors…

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