A conjecture of Borodin and a coloring of Grünbaum

We study the degenerate, the star and the degenerate star chromatic numbers and their relation to the genus of graphs. As a tool we prove the following strengthening of a result of Fertin et al. [8]: If G is a graph of maximum degree ∆, then G admits a degenerate star coloring using O(∆ 3/2 ) colors. We use this result to prove that every graph of genus g admits a degenerate star coloring with O(g 3/5 ) colors. It is also shown that these results are sharp up to a logarithmic factor.

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A conjecture of Borodin and a coloring of Grünbaum

Abstract: Abstract:In 1976, Borodin conjectured that every planar graph has a 5-coloring such that the union of every k color classes with 1 ≤ k ≤ 4 induces a (k − 1)-degenerate graph. We prove the existence of such a coloring using 18 colors.

Cited by 3 publications

References 14 publications

Improved upper bound for the degenerate and star chromatic numbers of graphs

Improved upper bound for the degenerate and star chromatic numbers of graphs

Colorings of plane graphs: A survey

Degenerate and star colorings of graphs on surfaces

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