Cooperative colorings of trees and of bipartite graphs

Given a system $\mathcal{G}=(G_1, \ldots ,G_m)$ of graphs on the same vertex set $V$, a cooperative coloring for $\mathcal{G}$ is a choice of vertex sets $I_1, \ldots ,I_m$, such that $I_j$ is independent in $G_j$ and $\bigcup_{j=1}^{m}I_j = V$. We give bounds on the minimal $m$ such that every $m$ graphs with maximum degree $d$ have a cooperative coloring, in the cases that (a) the graphs are trees, (b) the graphs are all bipartite.


Introduction
A set of vertices in a graph is called independent if no two vertices in it form an edge. A coloring of a graph G is a covering of V (G) by independent sets. Given a system G = (G 1 , . . . , G m ) of graphs on the same vertex set V , a cooperative coloring for G is a choice of vertex sets {I j ⊂ V : j ∈ [m]} such that I j is independent in G j and m j=1 I j = V . If all G j 's are the same graph G, then a cooperative coloring is just a coloring of G by m independent sets.
A basic fact about vertex coloring is that every graph G of maximum degree d is (d+1)-colorable. It is therefore natural to ask whether d+1 graphs, each of maximum degree d, always have a cooperative coloring. This was shown to be false: Theorem 1 (Theorem 5.1 of Aharoni, Holzman, Howard and Sprüssel [AHHS15]). For every d ≥ 2, there exist d + 1 graphs of maximum degree d that do not have a cooperative coloring.
As a cooperative coloring can be translated to an independent transversal (see [AHHS15, Section 2] for the connection), the fundamental result on independent transversals due to Haxell [Hax01, Theorem 2] implies that 2d graphs of maximum degree d always have a cooperative coloring. Let m(d) be the minimal m such that every m graphs of maximum degree d have a cooperative coloring. By the above, m(1) = 2 and (1) For example, the following was proved: Theorem 2 (Corollary 3.3 of Aharoni et al. [ABZ07] and Theorem 6.6 of Aharoni et al. [AHHS15]). Let C be the class of chordal graphs and let P be the class of paths. Then m C (d) = d + 1 for all d, and m P (2) = 3.
In this paper, we prove some bounds on m G (d) for another two classes: Theorem 3. Let T be the class of trees, and let B be the class of bipartite graphs. Then for d ≥ 2, log 2 log 2 d ≤m T (d) ≤ (1 + o(1)) log 4/3 d, Remark 1. Let F be the class of forests. It is evident that m F (d) ≥ m T (d) as F ⊃ T . Conversely, when d ≥ 2, given m = m T (d) forests F 1 , . . . , F m of maximum degree d, we can add a few edges to F i to obtain a tree F ′ i of maximum degree d, and the cooperative coloring for F ′ 1 , . . . , F ′ m is also a cooperative coloring for F 1 , . . . , F m . Therefore m F (d) = m T (d) for d ≥ 2.

Proof of Theorem 3 for trees
Proof of the lower bound on m T (d). Note that the system T 2 , consisting of the following two paths (one in thin red, the other in bold blue) does not have a cooperative coloring.
Suppose now that S = (F 1 , F 2 , . . . , F m ) is a system of forests on a vertex set V , not having a cooperative coloring. We shall construct a system Q(S) of m + 1 new forests F ′ 1 , F ′ 2 , . . . , F ′ m , F ′ m+1 , again not having a cooperative coloring.

The vertex set common to the new forests is
To these we add the (m + 1)st forest F ′ m+1 obtained by joining (z, u) to (u, v) for all u, v ∈ V . Assume that there is a cooperative coloring (I 1 , I 2 , . . . , I m , I m+1 ) for the system Q(S). Since the , this is contrary to our assumption that I m+1 is independent.
Note that |V ′ | = |V | 2 + |V | ≤ 2 |V | 2 . Note also that the maximum degree of Q(S) is attained in F ′ m+1 , and it is equal to |V |. Recursively define the system T m := Q(T m−1 ) consisting of m forets for m ≥ 3. Because the base T 2 has 4 vertices, one can check inductively that |V (T m )| is at most 2 3·2 m−2 −1 using |V (T m )| ≤ 2 |V (T m−1 )| 2 . Thus the maximum degree of T m is at most 2 3·2 m−3 −1 ≤ 2 2 m−1 . Given the maximum degree d ≥ 2, choose m := ⌈log 2 log 2 d⌉. By the choice of m, the maximum degree of T m is at most 2 2 m−1 ≤ d. By adding a few edges between the leaves in each forest of T m , we can obtain a system of m trees of maximum degree d that does not have a cooperative coloring. This means m T (d) > m > log 2 log 2 d.
Proof of the upper bound on m T (d). Let (T 1 , T 2 , . . . , T m ) be a system of trees of maximum degree d. We shall find a cooperative coloring by a random construction if m ≥ (1 + o(1)) log 4/3 d.
Choose arbitrarily for each tree T i a root so that we can talk about the parent or a sibling of a vertex that is not the root of T i . For each T i , choose independently a random vertex set S i , in which each vertex is included in S i independently with probability 1/2. Set R i := {v ∈ S i : v is a root or the parent of v ∈ S i } .
Since among any two adjacent vertices in T i one is the parent of the other, R i is independent in T i .
We shall show that with positive probability the sets R i form a cooperative coloring. For each vertex v, let B v be the event that v ∈ m i=1 R i . If v is the root of T i , then Pr (v ∈ R i ) = 1/2; otherwise Pr (v ∈ R i ) = 1/4. In any case, Pr (v ∈ R i ) ≤ 3/4, and so Pr (B v ) ≤ (3/4) m . Notice that B v is only dependent on the events B u for u that is the parent, a sibling or a child of v in some T i . Since the degree of v is at most d, it follows that B v is dependent on at most m × 2d other events. By the Lovász local lemma, if 3 4 then with positive probability no B v occurs, meaning that the sets R i form a cooperative coloring. The inequality (2) indeed holds under the assumption that m ≥ (1 + o(1)) log 4/3 d.

Proof of Theorem for bipartite graphs
Proof of the lower bound on m B (d). Given d, take m = ⌈log 2 d⌉. Let the vertex set be {0, 1} m , and for j ∈ [m] let G j be the complete bipartite graph between V 0 j and V 1 j where Note that the degree of G j is 2 m−1 ≤ d.
Suppose that I 1 , . . . , I m are independent sets in G 1 , . . . , G m respectively. As each G j is a complete bipartite graph, I j ⊆ V (3a) Consider the following random process. 1. For each a ∈ A, choose j = j(a) ∈ J L (a) uniformly at random, and put a in the set I j .
2. For each b ∈ B, choose arbitrarily j ∈ J R (b) \ {j(a) : a ∈ A, (a, b) ∈ E(G j )} =: J ′ R (b) as long as it is possible, and put b in the set I j .
For any a, a ′ ∈ A ∩ I j , a, a ′ ∈ L j and so (a, a ′ ) ∈ G j . This means A ∩ I j is independent, and similarly B ∩ I j is independent. For any b ∈ B ∩ I j and (a, b) ∈ E(G j ), by the definition of J ′ R (b), j(a) = j and so a ∈ I j . Therefore I j is independent for all j ∈ [m].
To prove the existence of a cooperative coloring it suffices to show that J ′ R (b) is nonempty for all b ∈ B with positive probability. For a vertex b ∈ B, let E b be the contrary event, that is, the event that J ′ R (b) is empty. For a fixed b ∈ B, let us estimate from above the probability of E b . For every j ∈ J R (b), let E j be the event that j ∈ J ′ R (b), that is the vent that j(a) = j for some a ∈ A that is a neighbor of b in G j . For each a ∈ A that is a neighbor of b in G j , we have As there are at most d neighbors of b in G j , we have for sufficiently large d that We claim that the events E j , j ∈ J R (b), are negatively correlated. This is easier to see with the complementary eventsĒ j , j ∈ J R (b). We have to show that for any choice of indices j 1 , . . . , The eventĒ j 1 ∩Ē j 2 ∩ · · · ∩Ē jt means that for all a ∈ A if a is a neighbor of b in G j i then j(a) = j i . Then, for any j ∈ {j 1 , . . . , j t }, for those vertices a ∈ A that are neighbors of b in G j , knowing that j(a) = j i for certain i ∈ [t] increases the probability that j(a) = j, and therefore increases the probability of E j . By the claim, the inequality (4) and the fact that E b = j∈J R (b) E j , we have The event E b is dependent on at most md 2 other events E b ′ , since for such dependence to exist it is necessary that b ′ ∈ B is at distance at most 2 from b in some graph G j . Thus, by the Lovász local lemma, for the positive probability that none of E b occurs it suffices that 1 d 4 × md 2 × e ≤ 1, which indeed holds for d sufficiently large as m = O(d).

Further directions
Cooperative colorings of graphs is a special case of a more general concept. Given a family H 1 , . . . , H t of hypergraphs, all sharing the same vertex set V , a cooperative cover is a choice of edges e i ∈ H i , such that i≤t e i = V . For a graph G let I(G) be the independence complex of G, namely the set