aa r X i v : . [ m a t h . C O ] S e p Asymptotically good edge correspondence colourings
Michael Molloy ∗ September 9, 2019
Abstract
We prove that every simple graph with maximum degree ∆ has edge correspondencenumber ∆ + o (∆). Graph colouring is one of the richest and most fundamental fields of graph theory. In itsmost basic form, one must assign a colour from a given set to each vertex of a graph so thatthe endpoints of each edge get different colours. Many variations have arisen, one of themost fruitful being list colouring: Vizing [29] and Erd˝os, Rubin Taylor [12] independentlysuggested that rather than assigning colours to all vertices from a single set, we can giveeach vertex v its own list of permissable colours, L ( v ). This very natural variation grew intoa prominent subfield of graph colouring.Recently, Dvo˘r´ak and Postle [10] introduced another natural variation, correspondencecolouring . Rather than using the same colouring rule for all edges, each edge can forbid adifferent set of pairs of colours on its endpoints. The only requirement is that no colour canbe forbidden to a vertex by two pairs on the same edge. Specifically, each edge uv is givena partial matching M uv between L ( u ) , L ( v ). The goal is to assign to each vertex v a colourfrom L ( v ) so that for every edge uv , the colours assigned to u and v are not paired in M uv .Note that if every edge uv simply matches each colour in L ( u ) ∩ L ( v ) to itself then we havethe usual list colouring. Several studies of correspondence colouring have already appeared,eg. [4, 5, 6, 7, 8, 13, 21]. See [7] for a discussion of how correspondence colouring can bemore challenging than list colouring, including how some common useful approaches to listcolouring do not apply to correspondence colouring.In an instance of correspondence colouring, if every list of colours has the same size, k ,then we can assume that each list is { , ..., k } . To see this, consider an instance where the ∗ Dept of Computer Science, University of Toronto, [email protected]. Research supported by anNSERC Discovery Grant. v take a bijection σ v : L ( v ) → { , ..., k } and for every edge uv ,replace each ( i, j ) ∈ M uv with ( σ v ( i ) , σ v ( j )). Similarly, when the lists have different sizes, wecan assume that the list of each vertex v is { , ..., | L ( v ) |} . So there is no difference betweene.g. the correspondence number and the list correspondence number of a graph.One of the most pursued open questions in list colouring is: When edge-colouring a simplegraph (i.e. assigning colours to the edges so that every two edges which share a vertex mustget different colours), are the identical lists the most difficult lists? In other words, is thelist edge chromatic number of a simple graph equal to the edge chromatic number? Thishas been answered in the affirmative for specific classes of graphs (eg. [14, 26]), but is stillopen for general graphs. In a seminal paper [18], Kahn proved that the two numbers areasymptotically equal: the list edge chromatic number of a simple graph with maximumdegree ∆ is equal to ∆ + o (∆). In a followup paper [19] he proved that the two numbers areasymptotically equal for multigraphs as well. Molloy and Reed [23] showed that, for simplegraphs, the o (∆) term is at most √ ∆poly(log ∆). See [16] for a more thorough backgroundto list colouring.Correspondence colouring can be defined for edge colouring in a natural way: each pairof edges that share a vertex is given a list of forbidden pairs (this is defined more formallybelow). Bernshteyn and Kostochka [7] showed that the edge correspondence number ofa simple graph can exceed the edge chromatic number. In fact, every ∆-regular simplegraph has edge correspondence number at least ∆ + 1, whereas many such graphs have edgechromatic number ∆. However, we show here that Kahn’s result holds in this context; i.e.every simple graph with maximum degree ∆ has edge correspondence number ∆ + o (∆).The previous best bound in this direction was (2 − ε )∆ for a constant ε >
0, which followsfrom work in [9] (in particular, the correspondence colouring version of their Theorem 1.6).To set things up formally: We are given a simple graph G and a set of colours Q = { , ..., q } . For each pair of incident edges e, f , we are given a partial matching M e,f on( Q , Q ); i.e. a collection of at most q pairs ( α, α ′ ) ∈ Q × Q such that each colour α is the firstelement of at most one pair and the second element of at most one pair. M f,e will consist ofthe reversal of all pairs in M e,f , so there is only one matching on each pair of incident edges.This collection of partial matchings is called an edge correspondence . An edge correspondencecolouring is an assignment to each edge e ∈ E ( G ) of a colour σ ( e ) ∈ Q , such that for everytwo incident edges e, f , the pair ( σ ( e ) , σ ( f )) is not in M e,f .The edge correspondence number of a graph G is the minimum q such that an edgecorrespondence colouring exists for every edge correspondence. We denote this by χ ′ DP ( G ),following the notation of Bernshteyn and Kostochka who refer to correspondence colouringas DP-colouring , using the intials of the founders.
Theorem 1.
Let G be any simple graph with maximum degree ∆ . Then χ ′ DP ( G ) = ∆+ o (∆) . Remark:
Throughout the paper, asymptotic notation is with respect to ∆ → ∞ .We colour the edges using an iterative procedure, first introduced in Kahn’s proof [18]and since then adapted to a very large number of results (see eg. [24]). At each step, we2olour a small proportion (roughly ) of the edges. We do so by considering a randomcolour assignment to those edges. If we name a particular vertex, then a probabilistic analysisshows that the colours of the edges near that vertex will likely satisfy certain properties, forexample that each edge has many remaining colours that can still be legally assigned to it.We apply the Lovasz Local Lemma to obtain a colouring in which the colours of the edgesnear every vertex satisfies those properties. Eventually we will have coloured almost all theedges; the remaining edges will be such that they are easily dealt with.One useful aspect to our procedure: we carry out the random colouring so that for anyedge e = uv , the effect of random choices involving edges incident to u is independent ofthe effect of choices on edges incident to v , and we track the cummulative affects of thosechoices seperately. This is where we make critical use of the facts: (i) G is simple, and (ii)edge colouring has the nice structural property that the neighbourhood of each edge consistsof two disjoint cliques. Let ε be any sufficiently small constant. We will prove that there exists ∆( ε ) such that if∆( G ) ≥ ∆( ε ) then χ ′ DP ( G ) ≤ (1 + ε )∆( G ). This is enough to establish Theorem 1. Wedo not name ∆( ε ) explicitly; instead we just assume that ∆( G ) is large enough to satisfyvarious inequalities that depend on ε .So we are given a graph G with maximum degree ∆, colours Q = { , ..., (1 + ε )∆ } andan edge correspondence. Our goal is to prove that, so long as ∆ is sufficiently large in termsof ε , there must be an edge correspondence colouring. We often use the following straightforward bound: (cid:18) ab (cid:19) ≤ (cid:16) eab (cid:17) b . We also rely on the following standard tool of the probabilistic method.
The Lov´asz Local Lemma [11].
Let A = { A , ..., A n } be a set of random events suchthat for each ≤ i ≤ n :(i) Pr ( A i ) ≤ p ; and(ii) A i is mutually independent of all but at most d other events. f pd ≤ then Pr ( A ∩ ... ∩ A n ) > . BIN ( n, p ) is the sum of n independent random boolean variables where each is equalto 1 with probability p . The following is a simplified special case of Chernoff’s bound. Itfollows from, e.g. Corollary A.1.10 and Theorem A.1.13 from Appendix A of [3]. The Chernoff Bound.
For any < t ≤ np : Pr ( | BIN ( n, p ) − np | > t ) < e − t / np . Theorem 2.3 from [22] generalizes the Chernoff Bound. Parts (b,c) of that theorem imply:
Lemma 2.
Suppose that we have independent random variables Z , ..., Z n , with ≤ Z i ≤ for each i . Set Z = P ni =1 Z i . For any < t ≤ E ( Z ) : Pr ( | Z − E ( Z ) | > t ) < e − t / E ( Z ) . Our final concentration tool is Talagrand’s Inequality, which often provides a strongerbound when the expectation of a random variable is much smaller than the number of trialsthat determine it. The following reworking of Talagrand’s original statement from [28], wasproved in the appendix of [25] . Talagrand’s Inequality.
Let X be a non-negative random variable determined by theindependent trials T , ..., T n . Suppose that for every set of possible outcomes of the trials, wehave:(i) changing the outcome of any one trial can affect X by at most ℓ ; and(ii) for each s > , if X ≥ s then there is a set of at most rs trials whose outcomes certifythat X ≥ s .Then for any t ≥ we have Pr ( | X − E ( X ) | > t + 20 ℓ p r E ( X ) + 64 ℓ r ) ≤ e − t ℓ r ( E ( X )+ t ) . We will find an edge correspondence colouring of the given graph using a common randomizedprocedure. One feature of that procedure is that when a edge gets a colour, any conflictingcolours are removed from the lists of available colours for all neighbouring edges. A similar statement appears as Talagrand’s Inequality II in [24]. Regretfully, there is an error in theproof of that statement and so we use this one instead. This is also discussed in [20]
4e would like to have applied the argument from [23] to prove that χ ′ DP ( G ) = ∆ + √ ∆poly(log ∆). The hurdle we could not overcome is as follows: The procedure in [23]begins by reserving a set of colours at each vertex which cannot be assigned to any edgesincident to that vertex. In the context of list edge colouring, this ensures that at the end ofthe procedure, each uncoloured edge uv can be assigned any of the colours that were reservedat both u and v ; however, this is not true for correspondence colouring.So instead we followed what is, at heart, the argument from [18], although presented asin [23, 24]. One difference is as follows: in that argument, one kept track of a parameter T ( v, c ) which was the set of edges incident to v which could still receive the colour c . Thatparameter was important because T ( u, c ) and T ( v, c ) comprised the edges which could cause c to be removed from the list of the edge e = uv . In the context of correspondence colouring,we need to redefine that parameter. For each edge e = uv we define T ( e, v, c ) to be the setof edges incident to v which can still receive a colour that will cause c to be removed fromthe list of e . Once that parameter is defined, the remainder of the argument is a simpleadaptation of those from [18, 23]. The following result is a very simple variation on the main result of [27] (improved in [15]).It will be used at the end of our proof, just as the result of [27] is used at the end of manysimilar proofs.As mentioned above, one can assume that each edge has the same list of permissiblecolours. Nevertheless, it will be convenient to extend the definition of an edge correspon-dence, and an edge correspondence colouring in the obvious way to the case where the listsmay differ.We use f ∼ e to denote that edges f, e are adjacent. For a pair of incident edges e, f , wesay that α ∈ L ( e ) has a partner in M e,f if α is the first element of one of the pairs in M e,f ;i.e. if assigning α to e forbids a colour to be assigned to f . Lemma 3.
We are given a simple graph G ; a list L ( e ) of size at least L on each edge e ;and an edge correspondence such that for each edge e and colour α ∈ L ( e ) , there are atmost T edges f ∼ e such that α has a partner in M e,f . If L ≥ T then there is an edgecorrespondence colouring. The proof is essentially identical to that from [27]; we include it here for completeness.It also follows easily from Theorem 2 of [15], with the constant 8 improved to 2.
Proof
Assign to each edge e a uniformly random colour from L ( e ). For each pair ofincident edges e, f and pair of colours ( α, α ′ ) ∈ M e,f we define A e,f,α,α ′ to be the event that e is assigned α and f is assigned α ′ . The probability of each such event is at most 1 /L . Eachevent A e,f,α,α ′ is easily seen to be mutually independent of all events which do not involve e or f ; i.e. of all but at most 2 LT other events. Since L × LT ≤ , the Lov´asz Local5emma implies that with positive probability none of these events hold; i.e. we obtain anedge correspondence colouring. (cid:3) We colour the graph randomly through a series of iterations, as described in the introduction.Roughly speaking, at each iteration we colour a small proportion of the edges. When anedge receives a colour then we remove any conflicting colours from the lists of incident edges.If two incident edges receive conflicting colours then both are uncoloured. A few technicalclarifications:(a) When an edge e receives a colour then conflicting colours are removed from all incidentedges even if that colour is removed from e . This is often refered to as wasteful since somecolours are needlessly removed from lists. We do this because it simplifies the analysis.Furthermore, because such a small proportion of edges are coloured, a vanishing proportionof coloured edges have their colour removed. As a result, the number of colours removedneedlessly from each list is negligible.(b) We allow each edge to receive multiple colours. For each edge e and each colour c ∈ L ( e ), the current list for e , we assign c to e with probability 1 / ( | L ( e ) | ln ∆); the choiceof whether to assign c to e is independent of the choices for all other colours in L ( e ). Sothe probability that e gets at least one colour is roughly 1 / ln ∆. Making these assignmentsindependently simplifies the analysis. And the probability that e gets at least two colours is O (ln − ∆) which is small enough to be negligible. We believe that this technique was firstused by Johansson in [17].(c) It is very convenient if, at each iteration, all lists have the same size and the probabilitythat a colour c is removed from L ( e ) is the same for every c, e . We enforce this by truncatingsome lists and by carrying out so-called equalizing coin flips which round up the probabilityof a colour being removed from a list.Our procedure makes use of the parameters L i , T i , Eq i ( e, c ). They will be defined formallybelow, as their definitions will be more intuitive after reading the procedure. For now, themain things to understand are: (i) our analysis will enforce that at the beginning of eachiteration i , every edge e has | L ( e ) | ≥ L i ; (ii) Eq i ( e, c ) is the value required for the equalizingcoin flips described above.If, during a particular iteration, colour c is assigned to e (in Step 2(b.i)) and colour c isnot unassigned from e (in Step 2(b.ii) or Step 2(c)) then we say that e retains c . At anyiteration, an edge is considered uncoloured if it did not retain any colour during the previousiterations.Recall from Section 2.1 that Q = { , ..., (1 + ε )∆ } .1. Initialize for every edge e = uv and colour c : L ( e ) = Q , T ( e, v, c ) is the set of all edges6ncident to e at v .2. For each i ≥ L i < ∆ / , T i < ∆ / or L i > T i :(a) For every uncoloured edge e with | L ( e ) | > L i , remove | L ( e ) | − L i arbitrary coloursfrom L ( e ).(b) For every uncoloured edge e and every colour c ∈ L ( e ):i. assign c to e with probability 1 / ( L i ln ∆).ii. If c was assigned to e then for every f ∼ e , if there is a colour c ′ ∈ L ( f ) with c ′ : f blocking c : e thenA. remove c ′ from L ( f ); andB. if c ′ was assigned to f then unassign c ′ from f (c) For every colour c still in L ( e ), with probability 1 − Eq i ( e, c ): remove c from L ( e ),and if c was assigned to e then unassign c from e .When this procedure terminates, each edge that has is not uncoloured is given one of thecolours that it retained. We will argue that it will terminate because L i > T i which willimply that this partial edge correspondence colouring can be completed using Lemma 3.Consider an edge e = uv . As mentioned in the introduction, we wish to seperate therandom choices related to the effect on e of edges around v from the effect of the edges around f . So to carry out the choice in line 2(c) whether to keep c in L ( e ), we will in fact maketwo independent random coin flips F ( e, u, c ) , F ( e, v, c ), which return 1 with probabilitiesEq i ( e, u, c ) and Eq i ( e, v, c ), respectively. If either returns 0 then c is removed from L ( e ).(The values of those probabilities are specified below.)For each edge e = uv and colour c , we define the following sets at the beginning of step2 of iteration i , i.e. after the lists have been truncated: L i ( e ) = the set of colours remaining in the list on eT i ( e, v, c ) = the set of uncoloured edges f containing v for whichthere is a colour c ′ ∈ L i ( f ) such that c ′ : f blocks c : e .We will recursively define parameters L i , T i and enforce that for each iteration i : | L i ( e ) | = L i and | T i ( e, v, c ) | ≤ T i for every edge e , endpoint v of e and colour c ∈ L i ( e ).(1)Note that the first condition means that | L ( e ) | ≥ L i at the beginning of iteration i .Recalling that we wish to focus seperately on colours removed from L ( e ) because ofedges around u and those removed because of colours around v , we introduce the followingterminology: 7 efinition 4. For an edge e = uv with c ∈ L ( e ) . We say that L ( e ) loses c at v duringiteration i if either (a) some edge f with endpoint v is assigned a colour c ′ where c ′ : f blocks c : e or (b) the equalizing coin flip F ( e, v, c ) returns 0. Note that if e is assigned the colour c in step 2(b) then c is unassigned from e iff L ( e )loses c at u or L ( e ) loses c at v .Suppose that (1) holds at the beginning of step 2 of iteration i . Thus the probability thatno colour c ′ is assigned to an edge f = wv where c ′ : f blocks c : e , is (cid:16) − L i ln ∆ (cid:17) | T i ( e,v,c ) | ≥ (cid:16) − L i ln ∆ (cid:17) T i .This inspires us to defineKeep i = (cid:18) − L i ln ∆ (cid:19) T i Eq i ( e, u, c ) = Keep i / (cid:18) − L i ln ∆ (cid:19) | T ( e,u,c ) | Eq i ( e, v, c ) = Keep i / (cid:18) − L i ln ∆ (cid:19) | T ( e,v,c ) | Eq i ( e, c ) = Eq i ( e, u, c ) × Eq i ( e, v, c )So the probability that L ( e ) loses c at v during iteration i is exactly 1 − Keep i , and theevent that it loses c at v is independent of the event that it loses c at u (since the graph issimple).Thus, the probability that c remains in L ( e ) at the end of iteration i is exactly Keep i ,and so the expected number of such colours remaining on L ( e ) is L i × Keep i .We now turn our attention to T i +1 ( e, v, c ). We cannot show that this parameter isconcentrated because it is possible for the assignment of a single colour to some f ∼ e tocause T i +1 ( e, v, c ) to drop to ∅ . So instead, we focus on a related parameter which essentiallyremoves the influence of edges incident to v . T ′ i +1 ( e, v, c ) is defined to be the set of edges f = vw ∈ T i ( e, v, c ) such that (a) f does notretain a colour during iteration i and (b) L ( f ) does not lose c ′ at w during iteration i , where c ′ is the unique colour in L ( f ) such that c ′ : f blocks c : e .Note that T i +1 ( e, v, c ) ⊆ T ′ i +1 ( e, v, c ). So an upper bound on | T ′ i +1 ( e, v, c ) | will provide anupper bound on | T i +1 ( e, v, c ) | . The fact that each colour in the list of an edge is assigned tothat edge independently, makes it simple to bound the expectation of | T ′ i +1 ( e, v, c ) | :For any edge f = vw and any colour α ∈ L i ( f ), let Z ( α, f ) be the event that α is assignedto f and let Y v ( α, f ) , Y w ( α, f ) be the events that L ( f ) loses α at v , and L ( f ) loses α at w during iteration i . The following observation is very helpful:8 bservation 5. The events { Z ( α, f ) , Y v ( α, f ) , Y w ( α, f ) : α ∈ L i ( f ) } are mutually inde-pendent. Proof
First, by the way we carry out Step 2(b), the events Z ( c, e ) over all edges e and c ∈ L i ( e ) are determined by independent trials. Y v ( α, f ) is determined by the events Z ( h, α ′ ) for all edges h ∈ T ( f, v, α ) and colours α ′ ∈ L ( g ) such that α ′ : h blocks α : f . Bythe nature of correspondence colouring, α ′ : h can block α : f for at most one colour α . Sincethe graph is simple, no edge h is relevant to both a Y v ( · , f ) event and a Y w ( · , f ) event. Sothese events are determined by disjoint sets of trials. (cid:3) Now consider any f ∈ T i ( e, v, c ), where c ′ : f blocks c : e . Suppose that (1) holds at thebeginning of iteration i . Then Observation 5 implies (see explanation below): Pr (cid:0) f ∈ T ′ i +1 ( e, v, c ) (cid:1) = Keep i × (cid:18) − L i Keep i (cid:19) × Y α ∈ L i ( f ) ,α = c ′ (cid:18) − L i Keep i (cid:19) < (cid:18) − L i Keep i (cid:19) L i since Keep i < < Keep i × (cid:18) − − ε/ i (cid:19) . Explanation:
The first term is the probability that L ( f ) does not lose c ′ at w . The secondterm is the probability that if f is assigned c ′ then L ( f ) loses c ′ at v and so c is removedfrom f . The third term is the probability that f does not retain any other colour.This yields that if (1) holds for iteration i then: E [ | T ′ i +1 ( e, v, c ) | ] < | T i ( e, v, c ) | × (cid:18) − − ε/ i (cid:19) × Keep i . (2)We will prove in section 4 that | T ′ i +1 ( e, v, c ) | and the number of colours removed from L ( e )during step 2 are both concentrated. This leads us to recursively define: L = (1 + ε )∆ , T =∆ and L i +1 = L i × Keep i − ∆ / (3) T i +1 = T i × (cid:18) − − ε/ i (cid:19) × Keep i + ∆ / . (4) Remark:
Recall that our procedure halts if L i or T i drops below ∆ / . It is not hardto show that Keep i = 1 − o (1) (see (5) below). So for all relevant values of i , L i is positiveand ∆ / is a second-order term in (3) and (4).We will prove: Lemma 6.
For every i ≥ , every edge e that is uncoloured at the beginning of iteration i ,each endpoint v of e , and every c ∈ L i ( e ) : if (1) holds for iteration i and L i , T i > ∆ / thenwith probability at least − ∆ − , at the beginning of iteration i + 1 we will have a) | L ( e ) | ≥ L i +1 ; and(b) | T ( e, v, c ) | ≤ T i +1 . The Lov´asz Local Lemma then implies that, with positive probability, the conditions ofLemma 6 hold simultaneously for every such e, c and so:
Lemma 7.
If (1) holds for iteration i and L i , T i > ∆ / then with positive probability (1)holds for iteration i + 1 . Proof
For each edge e = uv and colour c ∈ L i ( e ), we define A ( e ) to be the eventthat | L ( e ) | < L i +1 at the beginning of iteration i + 1, and B ( e, v, c ) to be the event that | T ( e, v, c ) | > T i +1 at the beginning of iteration i + 1. If none of these events hold, then (1)holds for iteration i + 1.Lemma 6 says that the probability of each such event is at most p := ∆ − . A ( e ) isdetermined by colour assignments and equalizing coin flips for edges incident with e ; B ( e, v, c )is determined by colour assignments and equalizing coin flips for edges within distance twoof e . So each event is mutually independent of all events involving edges at distance greaterthan four, and thus is mutually independent of all but at most d := 2∆ L i < ∆ other events(see e.g. the Mutual Independence Principle in [24]). Since pd < for large ∆, the LocalLemma completes the proof. (cid:3) A simple analysis of our recursive equations shows that T i decreases more quickly than L i , and so eventually their ratio will be large enough to allow us to apply Lemma 3. Wemust show that this happens before L i < ∆ / , as our procedure stops running if L i dropsbelow this value. Lemma 8.
For every sufficiently small ε > , there is an X = X ( ε ) such that for I = X ln ∆ we have L I > T I and L I , T I > ∆ / . Proof
Note that L /T = 1 + ε . We will prove inductively that L i /T i increases with i . Our first useful bound is: If L i /T i ≥ ε then:1 ≥ Keep i ≥ − T i L i ln ∆ > − ε ) ln ∆ . (5)Therefore for any constant X and i ≤ X ln ∆, if L i /T i ≥ ε and L j , T j ≥ ∆ / for all0 ≤ j ≤ i then: L i > T i > T i − Y j =1 (cid:18) − − ε/ j (cid:19) Keep j > T (cid:18) − (cid:19) I (cid:18) − ε ) ln ∆ (cid:19) I > ∆ e − X > ∆ / , (6)10or ∆ sufficiently large in terms of X, ε .We will prove that for any constant X , if for all 0 ≤ j ≤ i ≤ X ln ∆ we have L j /T j ≥ ε and L j , T j ≥ ∆ / then L i +1 T i +1 ≥ L i T i × (cid:16) ε (cid:17) , (7)for ∆ sufficiently large in terms of X, ε . It follows inductively that for all 1 ≤ i ≤ I = X ln ∆we have L i /T i ≥ ε and, by (6), L i , T i ≥ ∆ / . So the bound in (7) holds for all1 ≤ i ≤ I = X ln ∆.To prove (7), we first establish bounds on our recursive equations for L i , T i . The assump-tions that L i /T i > ε (and so (5) holds) and L i , T i > ∆ / imply: L i +1 = L i × Keep i − ∆ / > L i × Keep i (1 − ∆ − / ) (8) T i +1 = T i × (cid:18) − − ε/ i (cid:19) × Keep i + ∆ / < T i × (cid:18) − − ε/ i (cid:19) × Keep i (1 + ∆ − / ) . (9)Therefore L i +1 T i +1 ≥ L i T i × Keep i − − ε/ Keep i × − ∆ − / − / > L i T i × (cid:18) − ε ) ln ∆ (cid:19) × (cid:18) − ε/ i (cid:19) × (cid:0) − − / (cid:1) > L i T i × (cid:18) − (1 − ε + ε ) 1ln ∆ (cid:19) × (cid:18) − ε/ (cid:19) by (5) > L i T i × (cid:16) ε (cid:17) , for ε < and ∆ sufficiently large. This establishes (7). Therefore, if I = X ln ∆ where X is a constant that is sufficiently large in terms of ε , L I T I > L T × (cid:16) ε (cid:17) I > (1 + ε ) × (cid:16) ε (cid:17) X > . This and (6) prove the lemma. (cid:3)
Our main theorem follows immediately:
Proof of Theorem 1:
Setting I = X ln ∆ as in Lemma 8, (6) says that T I > ∆ / for∆ sufficiently large in terms of ε , so our procedure runs for at least I iterations. Since L i ≥ T i , and by the looping rule of our procedure, we have L i , T i > ∆ / at every iteration.So Lemma 7 shows inductively that with positive probability (1) holds at the beginning ofevery iteration. Thus Lemma 8 and the fact that L i /T i is increasing (as shown in the proof11f Lemma 8) yields that with positive probability, when the algorithm terminates, we willhave | L ( e ) | ≥ L i and | T ( e, v, c ) | ≤ T i for every uncoloured edge e , endpoint v of e and colour c ∈ L ( e ) where L i > T i . Now Lemma 3 shows that we can complete the colouring.This establishes that for every ε >
0, there exists ∆( ε ) such that every graph of maximumdegree ∆ ≥ ∆( e ) has edge correspondence number at most (1 + ε )∆. This implies our maintheorem. (cid:3) In this section, we prove our concentration lemma:
Proof of Lemma 6:
Part (a):
For any colour c remaining in L ( e ) after step 2(a) of iteration i , the probabilitythat c is not removed from L ( e ) during the remaining steps of iteration i is exactly Keep i ,as explained in section 3. Observation 1:
The event that c is not removed from L ( e ) is mutually independent ofthe corresponding events for any other colours of L ( e ).This follows immediately from: (i) for every edge f ∼ e and c ′ ∈ L ( f ) there is at mostone c ∈ L ( e ) such that c ′ : f blocks c : e , and (ii) whether c ′ is assigned to f is independentof the choice to assign any other colour to f or to assign any colour to any other edge.So the number of colours remaining after those steps is distributed like Bin( L i , Keep i ).Our hypothesis states L i > ∆ / and we know Keep i = 1 − o (1) by (5). So the ChernoffBounds imply the probability that fewer than L i +1 = L i × Keep i − ∆ / colours remain isat most 2 e − ∆ / / L i Keep i < ∆ − , for large ∆ since L i <
2∆ and Keep i < Part (b):
Let Ω be the set of edges f ∈ T i ( e, v, c ) that are not assigned any coloursduring interation i . Let Ω be the set of edges f ∈ T i ( e, v, c ) that are assigned at least onecolour and fewer than ∆ / colours during interation i . The expected number of edges in T i ( e, v, c ) \ (Ω ∪ Ω ) is at most T i (cid:18) L i ∆ / (cid:19) (cid:18) L i ln ∆ (cid:19) ∆ / < T i (cid:16) e ∆ / ln ∆ (cid:17) ∆ / < ∆ − . So by Markov’s Inequality, Pr[Ω ∪ Ω = T i ( e, v, c )] < ∆ − . (10)Recall that an edge f = vw ∈ T i ( e, v, c ) is not in T ′ i +1 ( e, v, c ) if (a) f is assigned andkeeps a colour, or (b) L ( f ) loses c ′ at w , where c ′ is the unique colour in L ( f ) such that c ′ : f blocks c : e ; if (b) occurs then we say that f loses the colour blocking c : e .
12e define: X is the number of edges f ∈ Ω such that f loses the colour blocking c : e ; X is the number of edges f ∈ Ω such that all colours assigned to f are then removed from f ; X is the number of edges f ∈ Ω such that f loses the colour blocking c : e and allcolours assigned to f are then removed from f . Note that if Ω ∪ Ω = T i ( e, v, c ) then | T ′ i +1 ( e, v, c ) | = | Ω | − X + X − X . (11)We will prove that Ω , X , X , X are all concentrated around their means. Then (10)and (11) will imply part (b).We start with Ω . Each edge f ∈ T i ( e, v, c ) goes into Ω with probability p = (1 − L i ln ∆ ) L i ≈ − and independently of whether any other edges enter Ω . So the Chernoffbound yields: Pr (cid:18) || Ω | − | T i ( e, v, c ) | p | >
18 ∆ / (cid:19) < e − ∆ / / T i p < ∆ − , (12)for sufficiently large ∆.To prove that X , X , X are concentrated, we will expose the colour assignments andequalizing coin flips in two phases: Phase 1.
We expose the colour assignments to all edges incident to v . Phase 2.
We expose the colour assignments to all remaining edges, and carry out allequalizing coin flips.
We use the notation E j , Pr j to denote expectation and probability over the randomchoices made during Phase j = 1 , E , Pr denotes the expectation and probability over theentire colour assignment. Let Ψ denote the colour assignments made in Phase 1. The analysisof Phase 2 will be conditional on Ψ and so we are interested in the following variables, definedfor t = 0 , , u t = E ( X t ) . Each u t is determined by Ψ. Simple properties of conditional expectations imply that E ( u t ) = E ( X t ) . (13)For each edge f ∈ T i ( e, v, c ), let ρ ( f ) denote the number of colours assigned to e byΨ, and set ρ + ( f ) = ρ ( f ) if f is assigned the colour blocking c : e at f , ρ + ( f ) = ρ ( f ) + 1otherwise. By Observation 1 from part (a), we have: u = (1 − Keep i ) | Ω | (14) u = X f ∈ Ω (1 − Keep i ) ρ ( f ) (15) u = X f ∈ Ω (1 − Keep i ) ρ + ( f ) (16)13e now show that E ( u i ) is large for each i . Since we halt if L i > T i , we haveKeep i ≤ (cid:16) − T i ln ∆ (cid:17) T < −
111 ln ∆ , for ∆ sufficiently large. It is easy to show that foreach f ∈ Ω , with probability at least we have ρ ( f ) = 1 , ρ + ( f ) = 2. Using the calculationsfrom (12), this yields: E ( u ) ≥
111 ln ∆ E ( | Ω | ) >
120 ln ∆ T i , E ( u ) ≥
120 ln ∆ E ( | Ω | ) ≈
130 ln ∆ T i , E ( u ) ≥ ∆ E ( | Ω | ) ≈ ∆ T i . In each case, the expectation is at least ∆ / since we have T i ≥ ∆ / by (6).We analyze the colour assignments of Phase 1 to prove that for each t = 0 , , [ | u t − E ( u t ) | >
18 ∆ / ] < ∆ − . (17)For i = 0, this bound follows immediately from (12). For i = 1 ,
2, note that u i is thesum of | T i ( u, v, c ) | independent variables, each bounded between 0 and 1; for example, when i = 1 the random variable corresponding to f is equal to 0 if f / ∈ Ω and (1 − Keep i ) ρ ( f ) otherwise. So (17) follows from Lemma 2.We next analyse the colour assignments and equalizing flips of Phase 2 to prove that foreach i = 0 , ,
2: Pr [ | X i − E ( X i ) | >
18 ∆ / ] < ∆ − . (18)We will use Talagrand’s Inequality. Note first that the assignment of a colour α toany edge w w can only affect whether at most two edges, w v and w v , are counted by X , X , X . So each colour assignment affects each of these variables by at most ℓ = 2.Next, suppose that X ≥ s ; i.e. at least s different edges in Ω lose the colour blocking c : e .Then each of s edges loses the colour blocking c : e because of a colour assignment to at leastone incident edge not incident with v or because of an equalizing coin flip. So there are s colour assignments or flips which certify that X ≥ s . If X ≥ s then each of a set of s edgeshad every colour that was assigned to it removed; since each edge in Ω was assigned fewerthan ∆ / colours, there is a set of fewer than ∆ / s colour assignments or equalizing flipsthat certify X ≥ s . Finally, if X ≥ s then there are at most ∆ / s colour assignments orflips that certify X ≥ s - one showing that each of the s edges loses the colour blocking c : e and fewer than ∆ / showing that it lost all its assigned colours. Note that each of thesethree variables has expectation at most T i ≤ ∆ and so we can apply Talagrand’s Inequalitywith ℓ = 2 , r = ∆ / and t = ∆ / , noting that t + 20 ℓ p r E ( X i ) + 64 ℓ r < ∆ / toestablish (18) for each i = 0 , , | X i − E ( X i ) | >
18 ∆ / ] < e − ∆ / / · / (∆+ ∆ / ) < ∆ − . | T ′ i ( e, v, c ) | is concentratedaround E ( | T ′ i ( e, v, c ) | ), which is bounded by (2). T i +1 ( e, v, c ) ⊆ T ′ i +1 ( e, v, c ) and so T i +1 ( e, v, c ) ≤ T ′ i +1 ( e, v, c ). This yields the upper bound of part (b) for ∆ sufficiently large. (cid:3) We close by remarking that our main theorem also holds for linear k -uniform hypergraphs for k = O (1). I.e., for any constant k , and any hypergraph H where every hyperedge containsexactly k vertices, every pair of vertices lies in at most one hyperedge, and every vertex liesin at most ∆ hyperedges, we have χ ′ DP ( G ) = ∆ + o (∆). The proof is a very straightforwardadaptation of the proof of Theorem 1. We outline it here:We first remark that our definintion of χ ′ DP extends naturally to linear hypergraphs.In the statement of Lemma 3, the constant 8 changes to 4 k . So in the halting conditionof our procedure, L i > T i becomes L i > kT i , and the constant 10 is replaced with 5 k asappropriate throughout the proof.For each hyperedge e , we define T i ( e, v, c ) , T ′ i ( e, v, c ) , Eq i ( e, v, c ) for each of the k vertices v in e .Equalizing coinflips ensure that for every v ∈ e and c ∈ L ( e ), the probability that L ( e )loses c at v is exactly 1 − Keep i . So now the probability that c remains in L ( e ) is Keep ki . Soour recursive equation for L i becomes: L i +1 = L i × Keep ki − ∆ / . This time, T ′ i +1 ( e, v, c ) is the number of edges f ⊆ T i ( e, v, c ) such that during iteration i ,(a) f does not retain a colour and (b) L ( f ) does not lose c ′ at any of its k − vertices otherthan v , where c ′ is the unique colour in L ( f ) such that c ′ : f blocks c : e . So equation (2)becomes: E [ | T ′ i +1 ( e, v, c ) | ] < | T i ( e, v, c ) | × (cid:18) − − ε/ ki (cid:19) × Keep k − i . So our recursive equation for T i becomes: T i +1 = T i × (cid:18) − − ε/ ki (cid:19) × Keep k − i + ∆ / . The statement of Observation 5 is modified to include k Y -events. The fact that H islinear ensures that this Observation still holds.The proof of Lemma 6 changes only in very straightforward places; eg. in the definitionsof u , u we multiply the exponent by k − = 2 becomes ℓ = k . (The fact that H is linear is important in the place where we set ℓ = k .) The same is true for Lemma 7. The change in our recursive equations results in veryminor changes to the calculations in the proof of Lemma 8. Those three lemma statementsremain the same, except for changing 10 to 5 k in Lemma 8.This modified proof yields: Theorem 9.
Let H be any linear uniform graph with maximum degree ∆ . Then χ ′ DP ( H ) =∆ + o (∆) . Acknowledgement
I am grateful to Runrun Liu and two anonymous referees for providing many corrections andimprovements to the first draft. This research is supported by an NSERC Discovery Grant.
References [1] N. Alon.
Independence numbers of locally sparse graphs and a ramsey type problem.
Rand. Str. & Alg. (1996) 271 - 278.[2] N. Alon, M. Krivelevich, B. Sudakov. Coloring graphs with sparse neighborhoods.
J.Comb. Th. (B) (1999), 73 - 82.[3] N. Alon and J. Spencer, The Probabilistic Method. Wiley, New York (1992).[4] A. Bernshteyn. The asymptotic behavior of the correspondence chromatic number.
Disc.Math. (2016), 2680 - 2692.[5] A. Bernshteyn.
The Johansson-Molloy Theorem for DP-coloring.
Rand. Struc. & Alg.(to appear).[6] A. Bernshteyn and A. Kostochka.
Sharp Diracs theorem for DP-critical graphs.
J. GraphTh. (2018), 521 - 546.[7] A. Bernshteyn and A. Kostochka. On differences between DP-coloring and list coloring. arXiv:1705.04883 (2017).[8] A. Bernshteyn, A. Kostochka, X. Zhu.
DP-colorings of graphs with high chromatic num-ber.
Eur. J. Comb. (2017), 122 - 129.[9] M. Bonamy, T. Perrett and L. Postle. Colouring graphs with Sparse Neighbourhoods:Bounds and Applications. arXiv:1810.06704 (2018).1610] Z. Dvo˘r´ak and L. Postle.
List-coloring embedded graphs without cycles of lengths 4 to 8.
J. Comb. Th. (B) (2018), 38 - 54.[11] P. Erd˝os and L. Lov´asz.
Problems and Results on -Chromatic Hypergraphs and SomeRelated Questions. In: ‘Infinite and Finite Sets” (A. Hajnal et. al. Eds), Colloq. Math.Soc. J. Bolyai , North Holland, Amsterdam (1975), 609 - 627.[12] P. Erd˝os, A. Rubin and H. Taylor. Choosability in graphs.
Congr. Num. (1979),125 - 157.[13] F. Feder and P. Hell. Complexity of correspondence homomorphisms. preprintarXiv:1703.05881 (2017).[14] F. Galvin.
The list chromatic index of a bipartite multigraph.
J. Comb. Th. (B) (1995), 153 - 158.[15] P. Haxell. A note on vertex list colouring.
Comb., Prob. & Comp. (2001), 345 - 347.[16] T. Jensen and B. Toft. Graph Colouring Problems.
Wiley (1995).[17] A. Johansson,
Asymptotic choice number for triangle free graphs.
Unpublishedmanuscript (1996).[18] J. Kahn.
Asymptotically good list colorings.
J Comb. Th. (A) (1996), 1 - 59.[19] J. Kahn. Asymptotics of the list chromatic index for multigraphs.
Rand. Struc. & Alg. (2000), 117 - 156.[20] T. Kelly and L. Postle. A local epsilon version of Reed’s Conjecture.
Manuscript.[21] S. Kim and K. Ozekib.
A sufficient condition for DP-4-colorability.
Disc. Math. (2018), 1983 - 1986.[22] C. McDiarmid.
Concentration.
In: Probabilistic Methods for Algorithmic DiscreteMathematics, (Habib M., McDiarmid C., Ramirez-Alfonsin J., Reed B., Eds.), Springer(1998), 195 - 248.[23] M. Molloy and B. Reed.
Near-optimal list colourings.
Rand. Struc. & Alg. (2000),376 - 402.[24] M. Molloy and B. Reed. Graph Colouring and the Probabilistic Method . Springer (2002).[25] M. Molloy and B. Reed,
Colouring graphs when the number of colours is almost themaximum degree . J. Comb. Th.(B) (2014), 134 - 195.[26] D. Peterson and D. R. Woodall.
Edge-choosability in line-perfect multigraphs.
Disc.Math. (1999), 191-199. 1727] B. Reed.
The list colouring constants.
J. Graph Th. (1999), 149 - 153.[28] M.Talagrand, Concentration of measure and isoperimetric inequalities in product spaces.
Instutut Des Hautes Etudes Scientifiques, Publications Mathematiques , 73 - 205(1995).[29] V. Vizing. Vertex colorings with given colors. (in Russian), Diskret. Analiz.,29