Colourings, transversals and local sparsity

Motivated both by recently introduced forms of list colouring and by earlier work on independent transversals subject to a local sparsity condition, we use the semi-random method to prove the following result. For any function $\mu$ satisfying $\mu(d)=o(d)$ as $d\to\infty$, there is a function $\lambda$ satisfying $\lambda(d)=d+o(d)$ as $d\to\infty$ such that the following holds. For any graph $H$ and any partition of its vertices into parts of size at least $\lambda$ such that (a) for each part the average over its vertices of degree to other parts is at most $d$, and (b) the maximum degree from a vertex to some other part is at most $\mu$, there is guaranteed to be a transversal of the parts that forms an independent set of $H$. This is a common strengthening of two results of Loh and Sudakov (2007) and Molloy and Thron (2012), each of which in turn implies an earlier result of Reed and Sudakov (2002).


Introduction
Let H be a graph with vertex partition W 1 , . . . , W m . An independent transversal of H with respect to {W i } m i=1 is a collection {w i } m i=1 of independent vertices in H such that w i ∈ W i for each i ∈ {1, . . . , m}. Writing ∆(H) for the maximum degree of H, the following classic combinatorial question is essentially due to Erdős (see [4]). (A) What is the least Λ = Λ(d) such that, for every H and W 1 , . . . , W m as above satisfying moreover that ∆(H) ≤ d and |W i | ≥ Λ for every i ∈ {1, . . . , m}, there is an independent transversal of H with respect to {W i } m i=1 ? Independently, Alon [1] and Fellows [10] showed that Λ is linear in d, and later, in an acclaimed work, Haxell (see [12,13]) used topological methods to prove that Λ(d) ≤ 2d. In fact, Λ(d) = 2d for every d as certified by an elementary construction due to Szabó and Tardos [20]. Now let G be a loopless multigraph, and let L : V (G) → 2 V (H) define a vertex partition of V (H), i.e. {L(v)} v∈V (G) defines a collection of disjoint subsets of V (H) whose union comprises V (H). An independent transversal of H with respect to L is an independent transversal of H with respect to {L(v)} v∈V (G) . We may assume without loss of generality that H is a cover graph for G via L: if vv ′ / ∈ E(G) then the bipartite subgraph of H induced between L(v) and L(v ′ ) is empty. Viewed in this way, the independent transversals in H may be related to vertex-colourings of G, as we now discuss.
Any mapping L : V (G) → 2 Z + is called a list-assignment of G; a colouring φ of V (G) is called an L-colouring if φ(v) ∈ L(v) for any v ∈ V (G). The problem of finding proper L-colourings for various natural choices of G is another famous combinatorial problem known as list colouring [9,22]. From G and L as above, we may produce a cover graph H ℓ = H ℓ (G, L) for G as follows. For every v ∈ V (G), let L ℓ (v) = {(v, c)} c∈L(v) . Let V (H ℓ ) = ∪ v∈V (G) L ℓ (v) and define E(H ℓ ) by letting (v, c)(v ′ , c ′ ) ∈ E(H ℓ ) if and only if vv ′ = e for some e ∈ E(G) and c = c ′ ∈ L(v) ∩ L(v ′ ). Then independent transversals of H ℓ with respect to L ℓ are in one-to-one correspondence with proper L-colourings of G.
For H being a cover graph for G via L, we need the notion of maximum colour multiplicity µ L (H) of H with respect to L, which is given by If G is a graph with list-assignment L, then µ L ℓ (H ℓ (G, L)) ≤ 1. Note that it makes no difference to H ℓ whether G is a multigraph or the underlying simple graph. In 2005, Aharoni and Holzman (see [15]) asked Question A in the special case when H is a cover graph for G via L satisfying µ L (H) ≤ 1. In particular, what is the smallest Λ 1 = Λ 1 (d) such that, if H is a cover graph for G via L satisfying moreover that ∆(H) ≤ d, µ L (H) = 1, and |L(v)| ≥ Λ 1 for every v ∈ V (G), then H has an independent transversal with respect to L? Loh and Sudakov [15] resolved this problem asymptotically by showing that Λ 1 (d) = d + o(d) as d → ∞. Furthermore they proved the same result under the milder assumption that µ L (H) = o(d) as d → ∞. Since Λ ℓ (d) ≤ Λ 1 (d) always, this also generalizes the aforementioned result of Reed and Sudakov. This question can also be expressed in the framework of correspondence colouring [7] (also known as DP-colouring), a more general form of the list colouring problem that has recently captivated the graph colouring community. A correspondence-assignment for G is a pair (L, M ) where L is a listassignment for G and M = {M e } e∈E(G) where M e is a matching between ∈ M e . Note that an (L, M )-colouring is not necessarily a proper colouring of G. Given a correspondence-assignment (L, M ), we may produce a cover graph H DP = H DP (G, (L, M )) for G as follows. For every v ∈ V (G), let L DP (v) = {(v, c)} c∈L(v) , and let V (H DP ) = ∪ v∈V (G) L DP (v). Define E(H DP ) by letting (v, c)(v ′ , c ′ ) ∈ E(H DP ) if and only if vv ′ = e for some e ∈ E(G) such that (v, c)(v ′ , c ′ ) ∈ M e . Then independent transversals of H DP with respect to L DP are in one-to-one correspondence with (L, M )-colourings of G. Morover, if G is a simple graph, then µ LDP (H DP ) ≤ 1, and whenever H ′ is a cover graph for a simple graph Thus, asking Question A with respect to H DP for simple G is equivalent to asking what is Λ 1 .
Note that in the special case for which the matchings M e "recognise" the colours, i.e.
and e ∈ E(G), then H DP is equivalent to H ℓ . Note also that µ LDP (H DP ) is at most µ(G), the maximum multiplicity of an edge in G.

Bounded average colour degrees
In this work we consider Question A and the above narrative in a further strengthened form. For H being a cover graph for G via L, let us define the maximum average colour degree ∆ L (H) of H with respect to L by We remark that occasionally we will drop the subscripts in µ L (H) and ∆ L (H) when the context is clear. Motivated by a graph colouring problem, the following natural variation upon Question A was implicitly asked recently (in the alternative formulation of single-conflict chromatic number, which we discuss later) by Dvořák, Esperet, Ozeki and the first author [6].
(B) What is the least Λ ′ = Λ ′ (d) such that, for every H and L as above satisfying moreover that ∆ L (H) ≤ d and |L(v)| ≥ Λ ′ for every v ∈ V (G), there is an independent transversal of H with respect to L? Note that since ∆ L (H) ≤ ∆(H) always, we have Λ ′ (d) ≥ Λ(d) = 2d. It was already observed that Λ ′ (d) ≤ 4d [6,Proposition 5], and for convenience we restate this in Proposition 4 below.
We can also ask Question B in the context of list colouring and correspondence colouring for simple G as before, and our main result resolves both of these questions in a stronger form. More fully, our main result is an asymptotically optimal bound in Question B in the special case that µ L (H) is a vanishingly small fraction of ∆ L (H).
For every H being a cover graph for G via L satisfying there is an independent transversal of H with respect to L.
Note that since ∆ L (H) ≤ ∆(H) always, Theorem 1 is stronger than the theorem of Loh and Sudakov, and is thus also stronger than the result of Reed and Sudakov. It also implies a more recent result of Molloy and Thron [17] on adaptable choosability (which itself also implies the result of Reed and Sudakov), which we now explain.
The "single-conflict" version of correspondence colouring a multigraph G concerns correspondence-assignments (L, M ) for G where the matchings M e for e ∈ E(G) have size 1. Equivalently, it concerns the existence of independent transversals in a graph H where H is a cover graph for G via L such that for is the multiplicity of the edge vv ′ in G. More precisely, the single-conflict chromatic number of a multigraph G is the smallest k such that the following holds: for every correspondence-assignment (L, M ) satisfying |L(v)| ≥ k for every v ∈ V (G) and |M e | = 1 for every e ∈ E(G), there is an (L, M )-colouring of G. Importantly, in this case every v ∈ V (G) satisfies |L(v)| · ∆ L (H) = ∆(G). Thus, we could equivalently ask Question B for such graphs H and replace ∆ L (H) with ∆(G)/Λ ′ , and this is essentially the same as asking for the best bound on the single-conflict chromatic number of multigraphs of bounded maximum degree. If we restrict the question further to the case where G has a list-assignment L ′ such that H is isomorphic to a subgraph of H ℓ (G, L ′ ), then similarly we are asking for the best bound on the adaptable choosability of multigraphs of bounded maximum degree. The adaptable choosability of a multigraph G is the smallest k such that the following holds: for every correspondence-assignment (L, M ) satisfying |L(v)| ≥ k for every v ∈ V (G) and |M e | = 1 for every e ∈ E(G) and moreover (u, c)(v, c ′ ) ∈ M uv only if c = c ′ , there is an (L, M )-colouring of G. In this way, Molloy and Thron's bound on the adaptable choosability implies that if G is a graph with list-assignment L and H ⊆ H ℓ (G, L) satisfying , then H has an independent transversal with respect to L ℓ . In this case, we still have µ L ℓ (H) ≤ 1, so Theorem 1 generalizes this result by allowing H ⊆ H DP (G, (L, M )) for a correspondence-assignment (L, M ) satisfying µ LDP (H) = o(d).
As in [19], the proof of Theorem 1 proceeds through a semi-random procedure. We have additionally incorporated ideas from both [15] and [17] as well as modern concentration tools.

Structure of the paper
In the next section, we present the probabilistic tools we require for the proof. We give an outline of the two-phase procedure in Section 3. The bulk of the paper is devoted to the proof of the second, main phase of the procedure in Section 4. At the end of the paper, we discuss a handful of interesting problems for further study.

Probabilistic tools
We need several probabilistic tools. The first such is the Lovász Local Lemma.
The Lovász Local Lemma. Let p ∈ [0, 1) and A a finite set of events such that for every A ∈ A, (i) P [A] ≤ p, and (ii) A is mutually independent of a set of all but at most d other events in A. If 4pd ≤ 1, then the probability that none of the events in A occur is strictly positive.
When we apply this, each bad event in A is an event in which a certain random variable deviates significantly from its expectation.
The remainder of this section is devoted to providing general sufficient conditions for a random variable to be concentrated around its expectation with high probability. The first and most basic of these is the Chernoff Bound.
In particular, when X i are indicator variables (i.e. a i = 0 and b i = 1), we have The Chernoff Bound provides very tight concentration, but is limited in its applicability. A much more flexible concentration inequality is Talagrand's Inequality [21]. It can be cumbersome though, so many researchers have proved derivations of it more suitable for combinatorial applications. We use the following version from [16], see [14,Remark 1].
Theorem 2 (Molloy and Reed [16]). Let X be a non-negative random variable determined by the independent trials T 1 , . . . , T n . Suppose that for every set of possible outcomes of the trials, we have that • changing the outcome of any one trial can affect X by at most δ; and • for each s > 0, if X ≥ s, then there is a set of at most rs trials whose outcomes certify that X ≥ s. Then for any t ≥ 0 where t/2 ≥ 20δ + rE [X] + 64δ 2 r, we have We also need a more robust version due to Bruhn and Joos [5, Theorem 7.5], which applies as long as almost all outcomes satisfy the conditions of Theorem 2, that is, it takes into account a set of exceedingly unlikely exceptional outcomes.
We say a random variable has upward (s, δ)-certificates with respect to a set of exceptional outcomes Ω * if for every ω ∈ Ω \ Ω * and every t > 0, there exists an index set I of size at most s so that X(ω ′ ) ≥ X(ω) − t for any ω ′ ∈ Ω \ Ω * for which the restrictions ω| I and ω ′ | I differ in at most t/δ coordinates. [5]). Let ((Ω i , Σ i , P i )) be probability spaces, let (Ω, Σ, P) be their product space, and let Ω * ⊆ Ω be a set of exceptional outcomes. Let X : Ω → R be a non-negative random variable, and let M = max{sup X, 1}, and let δ ≥ 1. If P [Ω * ] ≤ M −2 and X has upward (s, δ)-certificates, then for t > 50δ √ s,

A two-phase semi-random procedure
Dvořák, Esperet, Ozeki and the first author observed [6, Proposition 5] using the Lovász Local Lemma that every multigraph of maximum degree ∆ has singleconflict chromatic number at most ⌈ e(2∆ − 1)⌉. This implies that for every H being a cover graph for G via L satisfying ∆ L (H) ≤ d and |L(v)| ≥ 2ed, there is an independent transversal of H with respect to L. They remarked that by using the Local Cut Lemma [2] instead of the Lovász Local Lemma, one can improve the bound ⌈ e(2∆ − 1)⌉ to 2 √ ∆. This translates as follows.
, then there is an independent transversal of H with respect to L. Proposition 4 suffices as the "finishing blow" in our proof of Theorem 1. We reduce Theorem 1 to Proposition 4 using a two-phase semi-random procedure. We note that the value of the constant 4 is unimportant: our reduction also holds if 4 is replaced by 4e.
The first phase reduces the problem from one in which µ L (H) = o(d) to one in which µ L (H) ≤ d 1/5 , and this phase is embodied by the following result.
Theorem 5. For every d 1 , ε > 0, there exists γ 0 , d 0 > 0 such that following holds for all γ < γ 0 and d > d 0 . For every H being a cover graph for G via L satisfying Without the requirement that ∆(H ′ ) ≤ d ′ log 1/2 d ′ , the proof of Theorem 5 can be obtained from the proof of [15, Theorem 3.1] with the following substitutions, letting V (G) = {v 1 , . . . , v r }: . . , L(v r ), and • "local degree" → µ L (H). Effectively, the main difference is that we use the maximum average colour degree ∆ L (H) instead of ∆ L (H), and we obtain the weaker conclusion  Since the proof of Theorem 5 so closely resembles the proofs of these other results, we defer it to the appendix. For most of what remains of the paper, we focus on the second phase of our semi-random procedure.
For convenience, we introduce some further notation. If H is a cover graph for G via L then we say that an (L, , that is, the graph induced by H on the φ-useable colours, It is important to notice that, if φ is a proper partial (L, H)-colouring and G−C has an (L φ uncol , H φ uncol )-colouring, then H has an independent transversal with respect to L.
In the second, main phase, we find a sequence of proper partial (L, H)colourings φ of G in which we gradually improve the ratio of |L(v)|/∆ L (H) from (1+ε) for each v ∈ V (G) to 4 for each φ-uncoloured vertex, at which point we can apply Proposition 4. The second phase is embodied by the following theorem.
Theorem 6. For every ε > 0 and every H being a cover graph for G via L satisfying We prove Theorem 6 in Section 4. Our proof of Theorem 6 incorporates ideas from both [15] and [17]. We find the partial colouring φ in several iterations. Each iteration slightly improves the ratio of |L(v)|/∆(H) without affecting the other parameters too much, so that we can proceed for Θ(log d) iterations. The main hurdle is that µ L (H) can be relatively large, which affects the concentration of our random variables, but we can overcome this difficulty using Theorem 3, the "exceptional outcomes" version of Talagrand's Inequality.
We conclude this section with a proof of Theorem 1 assuming Theorem 6.
Proof of Theorem 1. By Theorem 5, it suffices to prove for any ε > 0 that for sufficiently large d ′ , if we have a cover graph H ′ for G via L ′ satisfying there is an independent transversal of H ′ with respect to L ′ . This is because since H ′ is an induced subgraph of H, an independent transversal of H ′ with respect to L ′ must also be an independent transversal of H with respect to L. Now by Theorem 6, there is a proper partial ( By Proposition 4, there is an independent transversal of H ′′ with respect to L ′′ , or equivalently, G − C has an (L ′′ , H ′′ )-colouring. By combining an (L ′′ , H ′′ )colouring of G − C with φ, we obtain an (L ′ , H ′ )-colouring of G, so H ′ has an independent transversal with respect to L ′ , as required.

The main phase
We prove Theorem 6 by applying several iterations of the following lemma.
Lemma 7. For every ε > 0, the following holds for sufficiently large d. If H is a cover graph for G via L satisfying Before proving Lemma 7, we first show how iteration of this lemma yields Theorem 6. As demonstrated in the proof below, applying this lemma improves the ratio of list size to maximum average colour degree, since crucially Proof of Theorem 6. Let p = log −1 d, d 0 = d and Λ 0 = (1 + ε)d, and for each integer 0 ≤ i ≤ 12/(εp), let Note that for every integer 0 ≤ i ≤ 12/(εp), we have so we may always assume d i is large enough to apply Lemma 7. We may moreover assume d log for every integer 1 ≤ i ≤ 12/(εp) we have and moreover In particular, there exists an integer 1 It only remains to prove Lemma 7. For the remainder of this section, let ε > 0 and d sufficiently large, let p satisfy log −1 d ≥ p ≥ log −2 d, and let H be a cover graph for G via L satisfying the hypotheses of Lemma 7. Throughout the proof we assume that if vv ′ ∈ E(G), then there exists c ∈ L(v) and c ′ ∈ L(v ′ ) such that cc ′ ∈ E(H). Thus ∆(G) ≤ Λd log d. Even when we do not explicitly state it, we will always assume that d is sufficiently large for certain inequalities to hold.
We will analyse a random proper partial (L, H)-colouring and use the Lovász Local Lemma to show that with nonzero probability it satisfies the properties we desire. Let us now describe this random colouring.
A wasteful (L, H)-colouring is a pair (A, φ) where • A ⊆ V (G) is a set of activated vertices and • φ is a partial (L, H)-colouring of G with domain A. Note that if (A, φ) is a wasteful (L, H)-colouring, then φ is not necessarily proper. We let A col be the set of vertices v ∈ A with no neighbor u ∈ A such that φ(v)φ(u) ∈ E(H).
To prove Lemma 7, we find a wasteful colouring (A, φ) such that every v ∈ V (G) satisfies In this case, we show that φ| A col satisfies the conclusion of Lemma 7 where C = A col , H ′ = H φ − ∪ v∈A col L(v), and L ′ = L φ | V (G)\A col . We call the colouring wasteful because there may be φ| A col -useable colours not in H ′ that we do not use.
The wasteful random colouring procedure with activation probability p samples a wasteful (L, H)-colouring (A, φ) as follows: In the analysis of this procedure, it will be helpful to define the following random variables for each vertex v ∈ V (G) and c ∈ L(v): Most of the proof is devoted to bounding the expected values of these random variables and showing that they are concentrated around their expectation with high probability. For proving concentration, it will be convenient to assume that φ = ψ| A where ψ is a (not necessarily proper) (L, H)-colouring of G where ψ(v) ∈ L(v) is chosen uniformly at random for each v ∈ V (G). In this way, the random variables above are determined by 2|V (G)| independent random trials, half of which determine which vertices in G are in A and half of which determine which colour is assigned to each vertex of G by ψ, which allows us to apply Theorems 2 and 3.
Our first step is to bound the expected values of some of these random variables. To this end, let By convexity of the exponential function and Jensen's Inequality, for v ∈ V (G), and for every c ∈ L(v), we have Proof. First (2) follows from (1) and Linearity of Expectation.

Now we prove (3). By Linearity of Expectation, we have
Keep(u, c ′′ ), and (3) follows from the above equality combined with (1), since Λ = |L(u)| for every u ∈ V (G). Now we need to show that the random variables in Claim 8 are close to their expectation with high probability. We use Theorem 3, the exceptional outcomes version of Talagrand's Inequality. To that end, we define an exceptional outcome for each vertex and show that it is unlikely. First, for each vertex u ∈ V (G) and c ∈ L(u), we define The events Ω * v include more outcomes than is necessary, but it is simpler to define it as we have. Now we bound the probability of these exceptional events.
Proof. First we let u ∈ V (G) and c ∈ L(u) and bound the probability that #conflicts u,c is too large:

By applying the bound deg H (c)
i < e deg H (c) i i and using the fact that deg H (c) ≤ d log d ≤ Λ log d, the righthand side of the above inequality is at most Since each term in the sum is at most (e/ log d) log 2 d and there are at most d log d terms, it follows that Since ∆(G) ≤ Λd log d, there are at most 16d 4 log 2 d vertices u ∈ N 2 G (v). Thus, combining the above inequality with the Union Bound, we have as required.
Having bounded the probability of the exceptional outcomes, we can now prove concentration of the random variables in Claim 8.
and for every c ∈ L(v), we have Proof. First we prove (4). Since it suffices to show that #unuseable cols v,c is concentrated. To that end, we show that #unuseable cols v,c has upward (s, δ)-certificates with respect to Ω * = ∅ where s = 2d log d and δ = d 1/4 . Let (A, φ) be a wasteful colouring. We construct a bipartite subgraph F ⊆ H with bipartition (F 1 , F 2 ) called the certificate graph, as follows. First, let . For each such c, we choose one such pair u, c ′ arbitrarily to put in F 2 , and we add the edge between c ∈ F 1 and (u, c ′ ) ∈ F 2 to the certificate graph F . We let I index the trials determining that u ∈ A and φ(u) = c ′ for each (u, c ′ ) ∈ F 2 . Note that |I| ≤ 2|L(v)| ≤ s, as required.
and therefore #unuseable cols v has upward (s, δ)-certificates, as claimed. Now by Theorem 3 applied with t = d 5/6 , we have and (4) follows. Now we prove (5). Since it suffices to show that both #activated nbrs v,c and #uncoloured nbrs v,c are concentrated.
Since #activated nbrs v,c is simply the sum of deg H (c) indicator variables, by the Chernoff Bound, we have Now we claim that #uncoloured nbrs v,c has upward (s, δ)-certificates with respect to Ω * v,c where s = 4d log d and δ = d 1/4 log 2 d. To that end, let (A, φ) / ∈ Ω * v,c be a wasteful colouring. We construct an auxiliary bipartite graph F with bipartition (F 1 , F 2 ) called the certificate graph, as follows. First, let F 1 be the set of colours c ′ ∈ N H (c) where c ′ ∈ L(u) for some u ∈ A \ A col . For each c ′ ∈ F 1 , there is a vertex w ∈ V (G) with colour φ(w) ∈ N H (φ(u)) certifying that u ∈ A\A col . For each such c ′ , we choose one such colour φ(w) arbitrarily and put (φ(u), φ(w)) ∈ F 2 , and we add the edge between c ′ ∈ F 1 and (φ(u), φ(w)) ∈ F 2 to the certificate graph F . We let I index the trials determining that u, w ∈ A and determining φ(u) and φ(w). Note that |I| ≤ 4 deg H (c) ≤ 4d log d = s, as required.
At this point we could use the Lovász Local Lemma to prove a weaker form of Lemma 7 with ∆ in the place of ∆ and use this to obtain an arguably simpler proof of the result of Reed and Sudakov [19] generalised in two ways: to the setting of correspondence colouring and to the setting of multigraphs of bounded multiplicity. The main simplification in the proof is the use of Theorem 3, the exceptional outcomes version of Talagrand's Inequality, to prove concentration of #uncoloured nbrs.
However, in order to prove Lemma 7 itself, we need to show that the φavailable colours remaining for each vertex do not predominantly have much larger degree in H than the average. To that end, we introduce the following notation: • we say a colour c ∈ L(v) is relevant if deg H (c) ≥ d/ log 3 d, and • for each v ∈ V (G), we let L rel (v) be the set of relevant colours in L(v). For each v ∈ V (G), we also define the random variables Our aim is to prove that remaining cols ′ old deg v is concentrated for each vertex v, but first we show that relevant cols ′ lost deg v is concentrated.
If we change the outcome of a single trial, then relevant cols ′ lost deg v is most affected if we change the colour of a vertex u to a colour c such that |N H (c) ∩ L rel (v)| = µ(H) and moreover each colour c ′ ∈ N H (c) ∩ L rel (v) has degree ∆(H) = d log d in H. Thus, changing the outcome of a trial affects relevant cols ′ lost deg v by at most δ, as required.
If relevant cols ′ lost deg v (A, φ) ≥ s, then there is a set of at most s log 3 d/d colours in L rel (v)\L φ (v), and for each such colour c, there is a vertex u ∈ A such that c ∈ N H (φ(u)). Thus, the trials determining that u ∈ A and φ(u) certify that relevant cols ′ lost deg v (A, φ) ≥ s, and there are at most 2s log 3 d/d = rs of them, as required. Therefore by Theorem 2, Now we use Claim 11 to prove that remaining cols ′ old deg v is concentrated for every vertex v ∈ V (G).
and, for d sufficiently large, Proof. First we prove (9). By Linearity of Expectation, For this, first we have where in the last line we used (9) and the definition of relevant. We also have Combining (11)-(13), we have and (10) follows.
At last we have all the ingredients -Claims 8, 10 and 12-necessary to prove Lemma 7 via the Lovász Local Lemma, as follows. Recall that log −1 d ≥ p ≥ log −2 d.
We sample a wasteful (L, H)-colouring by way of the wasteful random colouring procedure with activation probability p, as described earlier. We define the following set of bad events for each vertex v ∈ V (G) and c ∈ L(v): , and Letting A be the union of all such bad events, note that each event in A is mutually independent of all but at most (Λd log d) 4 other events in A, by our assumption that ∆(G) ≤ Λd log d. By Claims 8 and 10, using the fact that p ≥ log −2 d, for every v ∈ V (G) we have and for every c ∈ L(v) we have Moreover by Claim 12, for every v ∈ V (G) we have Therefore by (14)- (16) and the Lovász Local Lemma, there is a wasteful colouring (A, φ) / ∈ A, for all sufficiently large d. Now we show that φ| A col satisfies the conclusion of Lemma 7 where H ′ = H φ − ∪ v∈A col L(v) and L ′ = L φ | V (G)\A col . Since p ≤ log −1 d and we assume d is sufficiently large, we may assume p is sufficiently small for certain inequalities to hold. Indeed, for small enough p, every vertex v ∈ V (H ′ ) satisfies as required, where we used Claim 8 and the fact that where we used the fact that (A, φ) / ∈ A v,c for any c ∈ L ′ (v). And since (A, φ) / ∈ A v and (A, φ) / ∈ A ′ v for any v ∈ V (G), we have Now (1 + p 5/4 )/(1 − p 5/4 ) < 1 + 3p 5/4 for small enough p, and the righthand side above is at most 1 − p 1+ε/4 d. Thus ∆ L ′ (H ′ ) ≤ 1 − p 1+ε/4 d, as desired.

Conclusion
We conclude with some perspectives for future research.
First it is conceivable that Theorem 1 could be strengthened further. For H being the cover graph for G via L, let us define maximum average colour multiplicity µ L (H) of H with respect to L by Note that µ L (H) ≤ µ L (H) always. We believe that the statement of Theorem 1 also holds with µ L (H) in the place of µ L (H). This would imply a conjecture of Loh and Sudakov [15, p. 917] in a stronger form: they posited this with ∆(H) instead of ∆ L (H).
Second Λ ′ in Question B in general lies between 2d and 4d, as noted in the introduction, but its sharper determination remains a tempting problem.
Last we contend that many questions on independent transversals and colourings in terms of ∆ L (H) instead of ∆(H) may give rise to interesting challenges. Indeed, this work was partially motivated by such a study in terms of graphs embeddable in surfaces of prescribed genus [6, Theorem 1].

Note added
During the preparation of this manuscript, we learned of the concurrent and independent work of Glock and Sudakov [11]. They also proved Theorem 1 with a similar method. In their proof, they provided a weaker form of our Theorem 6 and established a more efficient form of our Theorem 5. This demonstrates considerable slack in the method and suggests that further refinement could lead to new developments.
Glock and Sudakov's work also had differing underlying motivation, more from independent transversals than from graph colouring. They proved Theorem 1 as a means toward the solution of certain problems about independent transversals (of which we had been unaware), especially one due to Erdős, Gyárfás and Luczak [8], from a quarter of a century ago.
Because it is brief, we include this easy application for the benefit of the reader. Erdős, Gyárfás and Luczak [8] asked for the determination of f (k), the least n such that, for any graph H on nk vertices having a partition L into parts of size k such that each bipartite subgraph induced between two distinct parts has no more than one edge, there is guaranteed to be an independent transversal. They showed that k 2 /(2e) ≤ f (k) ≤ (1 + o(1))k 2 as k → ∞. Note that every H and L as above satisfies ∆ L (H) ≤ (n − 1)/k. For a lower bound on f (k), it suffices by Theorem 1 to choose a suitable n = n(k) satisfying that k ≥ (1 + o(1))(n − 1)/k as k → ∞. This yields f (k) ≥ (1 + o(1))k 2 as k → ∞, matching their original upper bound, and settling their problem asymptotically. provided d is sufficiently large. Each of these events is mutually independent of all but at most d 3 other events, so by the Lovász Local Lemma, we may assume from here on that none of the events A v,c or B v,c,u happen.
We also need the following proposition (similar to [17,Proposition 4.1]) to reduce the maximum colour degree by removing a negligible number of colours for each vertex. Proof of Theorem 5. Let H be a cover graph for G via L satisfying |L(v)| ≥ (1 + ε)d for all v ∈ V (G), ∆ L (H) ≤ d, and µ L (H) ≤ γd. We may assume ε < 1, and we assume γ −1 and d are sufficiently large for various inequalities to hold throughout the proof. By Proposition 14, there is an induced subgraph H 0 ⊆ H that is a cover graph for G via L 0 for some L 0 satisfying |L(v)| ≥ (1 + 9ε/10)d, ∆ L ′ (H ′ ) ≤ d, and ∆ L ′ (H ′ ) ≤ d log 1/2 d. We may assume γ −6/5 < d, or else H 0 and L 0 satisfy all the required properties with d ′ = d, and the result follows.