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. They observed that the Local Lemma implies the conjecture for digraphs of large enough minimum out-degree if, crucially, the maximum in-degree is bounded by a(n exponential) function of the minimum out-degree.
 Our goal in this paper is to develop alternative methods that allow the verification of the conjecture for natural, broad digraph classes, without any restriction on the in-degrees. Among others, we prove the conjecture 1) for digraphs with chromatic number at most $6$ or dichromatic number at most $3$, and thus for all planar digraphs; and 2) for digraphs with maximum out-degree at most $4$. The benchmark case of $r$-regular digraphs remains open for $r \in [5,143]$.
 Our inductive proofs depend on loaded inductive statements about precoloring extensions of list-colorings. This approach also gives rise to stronger conclusions, involving the choosability version of majority coloring.
 We also give further evidence towards the existence of majority-$3$-colorings by showing that every digraph has a fractional majority 3.9602-coloring. Moreover we show that every digraph with large enough minimum out-degree has a fractional majority $(2+\varepsilon)$-coloring.
Highlights
We denote by δ+(D), δ−(D), ∆+(D), ∆−(D) the minimum or maximum out- or in-degree of D, respectively, and let ∆(D) = max{d+(v) + d−(v)|v ∈ V (D)} denote the maximum degree in D
The validity of the conjecture for any odd regularity r − 1 implies it for the even regularity r. This is the consequence of the fact3 that for even r any r-regular digraph D contains a 1-regular spanning subgraph F and any 3-majority coloring of the (r − 1)-regular digraph D − F is a majority coloring of D
In the second theorem of the section we show that digraphs with sufficiently large minimum out-degree have fractional majority colorings with total weight arbitrarily close to 2
Summary
All the proofs use the Local Lemma for a random coloring and require some upper bound on the maximum in-degree in terms of the minimum out-degree (in order to control the number of ”bad” events that are adjacent to any fixed bad event in some dependency graph of the events) As it is the case in many related open problems on splitting/coloring digraphs with large minimum out-degree [2, 16, 4, 9], large maximum in-degrees seem to be outside the realm of any such probabilistic approach and it looks like it constitutes the main difficulty of the problem. In this paper our main motivation is to complement the existing results on digraphs with balanced in- and out-degrees, and provide approaches for natural, broad families of digraphs, without any restriction on the maximum in-degree
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
