Majority Colourings of Digraphs

We prove that every digraph has a vertex 4-colouring such that for each vertex $v$, at most half the out-neighbours of $v$ receive the same colour as $v$. We then obtain several results related to the conjecture obtained by replacing 4 by 3.

This conjecture would be best possible. For example, a majority colouring of an odd directed cycle is proper (since each vertex has out-degree 1), and therefore three colours are necessary.
There are examples with large outdegree as well. For odd k 1 and prime n ≫ k, let G be the directed graph with V (G) = {v 0 , . . . , v n−1 } where N + G (v i ) = {v i+1 , . . . , v i+k } and vertex indices are taken modulo n. Suppose that G has a majority 2-colouring. If some sequence v i , v i+1 , . . . , v i+k contains more than k+1 2 vertices of one colour, say red, and v i is the leftmost red vertex in this sequence, then more than k−1 2 out-neighbours of v i are red, which is not allowed. Thus each sequence v i , v i+1 , . . . , v i+k contains exactly k+1 2 vertices of each colour. This implies that v i and v i+k+1 receive the same colour, as otherwise the sequence v i+1 , . . . , v i+k+1 would contain more than k+1 2 vertices of the colour assigned to v i+k+1 . For all vertices v i and v j , if ℓ = j−i k+1 in the finite field Z n , then j = i + ℓ(k + 1) and v i , v i+(k+1) , v i+2(k+1) , . . . , v i+ℓ(k+1) = v j all receive the same colour. Thus all the vertices receive the same colour, which is a contradiction. Hence the claimed 2-colouring does not exist.
Note that being majority c-colourable is not closed under taking induced subgraphs. For example, let G be the digraph with V (G) = {a, b, c, d} and E(G) = {ab, bc, ca, cd}. Then G has a majority 2-colouring: colour a and c by 1 and colour b and d by 2. But the subdigraph induced by {a, b, c} is a directed 3-cycle, which has no majority 2-colouring.
The remainder of the paper takes a probabilistic approach to Conjecture 2, proving several results that provide evidence for Conjecture 2. A probabilistic approach is reasonable, since in a random 3-colouring, one would expect that a third of the out-neighbours of each vertex v receive the same colour as v. So one might hope that there is enough slack to prove that for every vertex v, at most half the out-neighbours of v receive the same colour as v. Section 2 proves Conjecture 2 for digraphs with very large minimum outdegree (at least logarithmic in the number of vertcies), and then for digraphs with large minimum outdegree (at least a constant) and not extremely large maximum indegree. Section 3 shows that large minimum outdegree (at least a constant) is sufficient to prove the existence of one of the colour classes in Conjecture 2. Section 4 discusses multi-colour generalisations of Conjecture 2.
Before proceeding, we mention some related topics in the literature: • For undirected graphs, the situation is much simpler. Lovász [4] proved that for every undirected graph G and integer k 1, there is a k-colouring of G such that every vertex v has at most 1 k deg(v) neighbours receiving the same colour as v. The proof is simple. Consider a k-colouring of G that minimises the number of monochromatic edges. Suppose that some vertex v coloured i has greater than 1 k deg(v) neighbours coloured i. Thus less than k−1 k deg(v) neighbours of v are not coloured i, and less than 1 k deg(v) neighbours of v receive some colour j = i. Thus, if v is recoloured j, then the number of monochromatic edges decreases. Hence no vertex v has greater than 1 k deg(v) neighbours with the same colour as v.
• Seymour [6] considered digraph colourings such that every non-sink vertex receives a colour different from some outneighbour, and proved that a strongly-connected digraph G admits a 2-colouring with this property if and only G has an even directed cycle. The proof shows that every digraph has such a 3-colouring, which we repeat here: We may assume that G is strongly connected. In particular, there are no sink vertices. Choose a maximal set X of vertices such that G[X] admits a 3-colouring where every vertex has a colour different from some outneighbour. Since any directed cycle admits such a colouring, X = ∅. If X = V (G), then choose an edge uv entering X and colour u different from the colour of v, contradicting the maximality of X. So X = V (G). (The same proof show two colours suffice if you start with an even cycle.) • Alon [1,2] posed the following problem: Is there a constant c such that every digraph with minimum outdegree at least c can be vertex-partitioned into two induced digraphs, one with minimum outdegree at least 2, and the other with minimum outdegree at least 1?
• Wood [8] proved the following edge-colouring variant of majority colourings: For every digraph G and integer k 2, there is a partition of E(G) into k acyclic subgraphs such that each vertex v of G has outdegree at most k−1 ⌉ is best possible, since in each acyclic subgraph at least one vertex has outdegree 0.

Large Outdegree
We now show that minimum outdegree at least logarithmic in the number of vertices is sufficient to guarantee a majority 3-colouring. All logarithms are natural. Theorem 3. Every graph G with n vertices and minimum outdegree δ > 72 log(3n) has a majority 3-colouring. Moreover, at most half the out-neighbours of each vertex receive the same colour.
Proof. Randomly and independently colour each vertex of G with one of three colours {1, 2, 3}.
is determined by d v independent trials and changing the outcome of any one trial changes X(v, c) by at most 1. By the simple concentration bound 1 ,

The expected number of events
where the last inequality holds since δ > 72 log(3n). Thus there exists colour choices such that no event A(v, c) holds. That is, a majority 3-colouring exists.
The following result shows that large outdegree (at least a constant) and not extremely large indegree is sufficient to guarantee a majority 3-colouring. Proof. We assume δ 1200, as otherwise the minimum out-degree δ is greater than the maximum in-degree exp(δ/72)/12δ, which does not make sense.
We use the following weighted version of the Local Lemma [3,5]: Assume there are numbers t 1 , . . . , t n 1 and a real number p ∈ [0, 1 4 ] such that for 1 i n, (a) P(A i ) p t i and (b) Then with positive probability no event A i occurs.
Define p := exp(−δ/72). Since δ 1200 we have p ∈ [0, 1 4 ]. Randomly and independently colour each vertex of G with one of three colours {1, 2, 3}. Consider a vertex v with out-degree d v . Let X(v, c) be the random variable that counts the number of out-neighbours of v coloured c. Note that X(v, c) is determined by d v independent trials and changing the outcome of any one trial changes X(v, c) by at most 1. By the simple concentration bound, Thus condition (a) is satisfied. For each event A(v, c) let D(v, c) be the set of all events A(w, c ′ ) ∈ A such that v and w have a common out-neighbour. Then A(v, c) is mutually Since each out-neighbour of v has in-degree at most exp(δ/72)/12δ, we have |D(v, c)| d v exp(δ/72)/4δ and Thus condition (b) is satisfied. By the local lemma, with positive probability, no event A(v, c) occurs. That is, a majority 3-colouring exists.
Note that the conclusion in Theorem 3 and Theorem 4 is stronger than in Conjecture 2. We now show that such a conclusion is impossible (without some extra degree assumption).

Lemma 5.
For all integers k and δ, there are infinitely many digraphs G with minimum outdegree δ, such that for every vertex k-colouring of G, there is a vertex v such that all the out-neighbours of v receive the same colour.
Proof. Start with a digraph G 0 with at least kδ vertices and minimum outdegree δ. For each set S of δ vertices in G 0 , add a new vertex with out-neighbourhood S. Let G be the digraph obtained. In every k-colouring of G, at least δ vertices in G 0 receive the same colour, which implies that for some vertex v ∈ V (G) \ V (G 0 ), all the out-neighbours of v receive the same colour.

Stable Sets
A set T of vertices in a digraph G is a stable set if for each vertex v ∈ T , at most half the out-neighbours of v are also in T . A majority colouring is a partition into stable sets. Of course, if a digraph has a majority 3-colouring, then it contains a stable set with at least one third of the vertices. The next lemma provides a sufficient condition for the existence of such a set. Theorem 6. Every digraph G with n vertices and minimum outdegree at least 22 has a stable set with at least n 3 vertices.
Theorem 6 is proved via the following more general lemma.

Lemma 7.
For 0 < α < p < β < 1, every digraph G with minimum outdegree at least For each vertex v of G, add v to S independently and randomly with probability p. Let By the Chernoff bound 2 , where the last inequality follows from the definition of δ.
Since the events v ∈ S and X v > βd v are independent, Let T := S \ B. Thus |N + G (v) ∩ T | βd v for each vertex v ∈ T , as desired. By the linearity of expectation, Thus there exists the desired set T . Note the following corollary of Lemma 7 obtained with α = 1 2 − ǫ and p = 1 2 − ǫ 2 . This says that graphs with large minimum outdegree have a stable set with close to half the vertices. Proposition 8. For 0 < ǫ < 1 2 , every n-vertex digraph G with minimum outdegree at least 2ǫ −2 (2 − ǫ) log( 1−ǫ ǫ ) contains a stable set of at least ( 1 2 − ǫ)n vertices.

Multi-Colour Generalisation
The following natural generalisation of Conjecture 2 arises.
Conjecture 9. For k 2, every digraph has a vertex (k + 1)-colouring such that for each vertex v, at most 1 k deg + (v) out-neighbours of v receive the same colour as v.
The proof of Theorem 1 generalises to give an upper bound of k 2 on the number of colours in Conjecture 9. It is open whether the number of colours is O(k). This conjecture would be best possible, as shown by the following example. Let G be the k-th power of an n-cycle, with arcs oriented clockwise, where n 2k + 3 and n ≡ 0 (mod k + 1). Each vertex has outdegree k. Say G has a vertex (k + 1)-colouring such that for each vertex v, at most ǫk out-neighbours of v receive the same colour as v. If ǫk < 1 then the underlying undirected graph of G is properly coloured, which is only possible if n ≡ 0 (mod k + 1). Hence ǫ 1 k .
Proposition 10. For k 2 and ǫ ∈ (0, 1 k ), every n-vertex digraph G with minimum outdegree at least 2ǫ −2 ( 4 k − ǫ) log 2 ǫk − 1 contains a set T of at least ( 1 k − ǫ)n vertices, such that for every vertex v ∈ T , at most 1 k deg + (v) out-neighbours of v are also in T .

Open Problems
In addition to resolving Conjecture 2, the following open problems arise from this paper: 1. Is there a constant β < 1 for which every digraph has a 3-colouring, such that for every vertex v, at most β deg + (v) out-neighbours receive the same colour as v?
2. Does every tournament have a majority 3-colouring?
3. Does every Eulerian digraph have a majority 3-colouring? Note that for an Eulerian digraph G, if each vertex v has in-degree and out-degree deg(v), then by the result for undirected graphs mentioned in Section 1, the underlying undirected graph of G has a 4-colouring such that each vertex v has at most 1 2 deg(v) in-or-out-neighbours with the same colour as v. In particular, G has a majority 4-colouring. By an analogous argument every Eulerian digraph has a 3-colouring such that each vertex v has at most 2 3 deg(v) in-or-out-neighbours with the same colour as v, thus proving a special case of the first question above. 4. Does every digraph in which every vertex has in-degree and out-degree k have a majority 3-colouring? A variant of Theorem 4 proves this result for k 144.

5.
Is there a characterisation of digraphs that have a majority 2-colouring (or a polynomial time algorithm to recognise such digraphs)?
6. Does every digraph have a O(k)-colouring such that for each vertex v, at most 1 k deg + (v) out-neighbours receive the same colour as v (for all k 2)? colouring is a function that assigns each stable set T ∈ S(G) a weight x T 0 such that T ∈S(G,v) x T 1 for each vertex v of G. What is the minimum number k such that every digraph G has a fractional majority colouring with total weight T ∈S(G) x T k? Perhaps it is less than 3.