• DOI:10.1002/jgt.22827
• Corpus ID: 221377259

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

@article{Bradshaw2020GraphCW,
  title={Graph colorings with restricted bicolored subgraphs: I. Acyclic, star, and treewidth colorings},
  author={Peter Bradshaw},
  journal={Journal of Graph Theory},
  year={2020},
  volume={100},
  pages={362 - 370}
}

• Peter Bradshaw
• 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 suffice, and acyclic colorings, for which O ( Δ 4 ∕ 3 ) $O({{\rm{\Delta }}}^{4\unicode{x02215}3})$ colors suffice. Our proof uses a probabilistic method of Alon, McDiarmid, and Reed. This result also gives previously unknown upper bounds, including the fact that a graph of maximum degree Δ ${\rm{\Delta }}$ has a proper coloring with O ( Δ 9 ∕ 8 ) $O({{\rm{\Delta }}}^{9\unicode{x02215}8})$ colors in which every bicolored subgraph is planar, as well as a proper coloring with O ( Δ 13 ∕ 12 ) $O({{\rm{\Delta }}}^{13\unicode{x02215}12})$ colors in which every bicolored subgraph has treewidth at most 3.