Majority Colorings of Sparse Digraphs

TL;DR

This paper investigates majority colorings in directed graphs, proving new bounds for list and fractional colorings, and resolving open questions about complexity and colorability thresholds.

Contribution

It establishes new bounds for majority 3-colorability in digraphs with certain chromatic and degree constraints, and explores fractional and list coloring variants.

Findings

Every digraph with chromatic number ≤ 6 is majority 3-colorable.

Deciding majority 2-colorability is NP-complete.

Every digraph has a fractional majority 3.9602-coloring.

Abstract

A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority 4-coloring and conjectured that every digraph admits a majority 3-coloring. We verify this conjecture for digraphs with chromatic number at most 6 or dichromatic number at most 3. We obtain analogous results for list coloring: We show that every digraph with list chromatic number at most 6 or list dichromatic number at most 3 is majority 3-choosable. We deduce that digraphs with maximum out-degree at most 4 or maximum degree at most 7 are majority 3-choosable. On the way to these results we investigate digraphs admitting a majority 2-coloring. We show that every digraph without odd directed cycles is majority 2-choosable. We answer an open question posed by Kreutzer et al. negatively, by showing that deciding whether a given digraph is majority 2-colorable is NP-complete. Finally we deal with a fractional relaxation of majority coloring proposed by Kreutzer et al. and show that every digraph has a fractional majority 3.9602-coloring. We show that every digraph with minimum out-degree $\Omega\left((1/\varepsilon)^2\ln(1/\varepsilon)\right)$ has a fractional majority $(2+\varepsilon)$-coloring.