Independent transversals in locally sparse graphs
aa r X i v : . [ m a t h . C O ] J un Independent transversals in locally sparse graphs
Po-Shen Loh ∗ Benny Sudakov † Abstract
Let G be a graph with maximum degree ∆ whose vertex set is partitioned into parts V ( G ) = V ∪ . . . ∪ V r . A transversal is a subset of V ( G ) containing exactly one vertex from each part V i . If itis also an independent set, then we call it an independent transversal. The local degree of G is themaximum number of neighbors of a vertex v in a part V i , taken over all choices of V i and v V i .We prove that for every fixed ǫ >
0, if all part sizes | V i | ≥ (1 + ǫ )∆ and the local degree of G is o (∆), then G has an independent transversal for sufficiently large ∆. This extends several previousresults and settles (in a stronger form) a conjecture of Aharoni and Holzman. We then generalizethis result to transversals that induce no cliques of size s . (Note that independent transversalscorrespond to s = 2.) In that context, we prove that parts of size | V i | ≥ (1 + ǫ ) ∆ s − and local degree o (∆) guarantee the existence of such a transversal, and we provide a construction that shows thisis asymptotically tight. Let G = ( V, E ) be a graph with maximum degree ∆, whose vertices have been partitioned into r disjointsets V = V ∪ . . . ∪ V r . An independent transversal of G with respect to { V i } ri =1 is an independent setin G which contains exactly one vertex from each V i . The problem of finding sufficient conditions forthe existence of an independent transversal dates back to 1975, when it was raised by Bollob´as, Erd˝os,and Szemer´edi [7]. Since then, much work has been done [1, 2, 3, 4, 5, 10, 11, 13, 14, 15, 18, 19, 20],and this basic concept has also appeared in the study of other combinatorial problems, such as lineararboricity, strong chromatic number and list coloring. In particular, as part of his result on the lineararboricity of graphs, Alon [4] used the Lov´asz Local Lemma to show that an independent transversalexists as long as all | V i | ≥ e ∆. Haxell [10] later improved his constant from 2 e to 2. In the otherdirection, Jin [14] and Yuster [20] constructed graphs with parts of size | V i | = 2∆ − − K ∆ , ∆ and the the partition into { V i } is done in such a way that the parts { V i } separate the sides of each K ∆ , ∆ . This creates many pairs of disjoint parts ( V i , V j ) which have complete ∗ Department of Mathematics, Princeton University, Princeton, NJ 08544. E-mail: [email protected]. Re-search supported in part by a Fannie and John Hertz Foundation Fellowship, an NSF Graduate Research Fellowship,and a Princeton Centennial Fellowship. † Department of Mathematics, Princeton University, Princeton, NJ 08544, and Institute for Advanced Study, Prince-ton. E-mail: [email protected]. Research supported in part by NSF CAREER award DMS-0546523, NSFgrant DMS-0355497, USA-Israeli BSF grant, Alfred P. Sloan fellowship, and the State of New Jersey. V i , V j ) is quadratic in ∆. In this paper we show that the constant 2 can be significantly improvedif one prohibits such phenomena. One way to accomplish this is to introduce a constraint on the localdegree , which is the maximum number of neighbors of a vertex v in a part V i , where V i ranges overall parts and v ranges over all vertices v V i . This constraint arises naturally in several contexts, oneof which is vertex list coloring.Given a graph H = ( V, E ) and a set of lists { C v } of available colors, one for each vertex v ∈ V ,it is a natural question to determine when we can properly color H from these lists. Suppose that inaddition we know that every color c appears in the lists of at most ∆ neighbors of each vertex v ; then,what minimum size lists will guarantee a proper coloring? This question, which was proposed by Reed[17], can be recast as an independent transversal problem as follows. Consider a | V | -partite graph G such that for each v ∈ V , G has a part with | C v | vertices labeled by ordered pairs { ( v, c ) : c ∈ C v } .Let two vertices ( v, c ) and ( w, c ) be adjacent whenever v is adjacent to w in H and c ∈ C v ∩ C w . Then G has maximum degree ≤ ∆ and local degree ≤
1, and an independent transversal in G correspondsto a proper list coloring of H . (Note that not every G with local degree 1 has a corresponding listcoloring problem, so this association is not reversible.) Haxell’s result immediately implies that if all | C v | ≥ o (1))∆ will suffice.Aharoni and Holzman [2] adapted arguments from [18] to prove the existence of an independenttransversal in multipartite graphs with maximum degree ∆, parts of size (1 + o (1))∆, and the propertythat any two distinct vertices in the same part are at distance greater than 4 from each other. Theirresult has the following nice application. For any collection of n ≥ (1 + o (1))∆ graphs { H i } ni =1 withmaximum degree ∆, all sharing the same vertex set V , there exists a partition V = S ni =1 I i such thatfor each i , I i is an independent set in H i . To see this, create a multipartite graph G by making n copies of each vertex, and connect the i -th copy of vertex v to the i -th copy of vertex w if v is adjacentto w in H i . Then in G there are no paths at all between any pair of distinct vertices in a given part.Thus we can find an independent transversal of G , which gives the required partition.Aharoni and Holzman [2] conjectured that their condition on distances could be replaced by theweaker condition that the local degree is 1. In this paper, we prove the following stronger theorem,which implies their conjecture. Our proof combines arguments from [5] and [18], together with someadditional ideas. Theorem 1.1
For every ǫ > there exists γ > such that the following holds. If G is a graph withmaximum degree at most ∆ whose vertex set is partitioned into parts V ( G ) = V ∪ . . . ∪ V r of size | V i | ≥ (1 + ǫ )∆ , and the local degree of G is at most γ ∆ , then G has an independent transversal. Note that the constant of 1 is optimal because of the following example: a disjoint union of ∆ cliquesof order ∆ + 1, where each clique has exactly one vertex per part.An independent transversal is a set with one vertex from each part V i that induces no cliquesof size 2. Therefore, a natural generalization of this concept is the K s -free transversal, which is a2ransversal inducing no cliques of size s . Such transversals were recently studied by Szab´o and Tardos[19], who posed the problem of finding p (∆ , K s ), which is defined to be the smallest integer n thatguarantees the existence of a K s -free transversal in any graph with maximum degree ∆ and part sizesat least n . They provided a construction that bounds p (∆ , K s ) ≥ ss − s − , and conjectured that theirconstruction was optimal.However, this construction also contains complete bipartite subgraphs of linear size, with sidesseparated by the partition. In light of our previous result, one may ask what can be said when weimpose a local degree restriction. We find that we can solve that problem asymptotically, and provethe following generalization of Theorem 1.1: Theorem 1.2
For every ǫ > and integer s ≥ , there exists γ > such that the following holds. If G is a graph with maximum degree at most ∆ whose vertex set is partitioned into parts V ( G ) = V ∪ . . . ∪ V r of size | V i | ≥ (1 + ǫ ) ∆ s − , and the local degree of G is at most γ ∆ , then G has a K s -free transversal. This is asymptotically tight via a simple construction that we will give later. Furthermore, a slightadaptation of our method proves that even without the local degree condition, p (∆ , K s ) ≤ (cid:4) ∆ s − (cid:5) ,which differs from Szab´o and Tardos’s conjecture by a factor of at most 2. For s = 3, this matchestheir best known upper bound for p (∆ , K ), and even is better by 1 when ∆ is odd.The rest of this paper is organized as follows. The next section reviews some basic probabilistictools we use in our proof. In Section 3 we show how to reduce Theorem 1.1 to the special casewhen local degrees are bounded by a constant. We solve this case in Section 4. In Section 5, weprove the generalization of our main result to K s -free transversals. The final section contains someconcluding remarks and open problems. Throughout this paper we will assume wherever needed that γ is sufficiently small. Since, by definition, every non-trivial r -partite graph has local degree at leastone, this implies that ∆ ≥ γ − is sufficiently large. We will also systematically omit floor and ceilingsigns for the sake of clarity of presentation. In this section we describe some classical results which we will use in our proof. We begin with severallarge-deviation inequalities.
Theorem 2.1 (Hoeffding [9], Chernoff [6]) Let X = P ni =1 X i be a sum of bounded independentrandom variables a i ≤ X i ≤ b i . Then if we let µ = E [ X ] , P [ | X − µ | ≥ t ] ≤ (cid:26) − t P ni =1 ( b i − a i ) (cid:27) . In particular, when X i are indicator variables we have P (cid:2) | X − µ | > t (cid:3) < e − t /n . Also, for any ǫ > , there exists c ǫ > such that P (cid:2) | X − µ | > ǫµ (cid:3) < e − c ǫ µ .
3o state the next concentration result, we need to introduce two concepts. Let Ω = Q ni =1 Ω i be aprobability space, and let X : Ω → R be a random variable. • Suppose that there is a constant C such that changing ω in any single coordinate affects thevalue of X ( ω ) by at most C . Then we say that X is C -Lipschitz . • Suppose that for every s and ω such that X ( ω ) ≥ s , there exists a set I ⊂ { , . . . , n } of size | I | ≤ rs such that every ω ′ that agrees with ω on the coordinates indexed by I also has X ( ω ′ ) ≥ s .Then we say that X is r -certifiable. Theorem 2.2 (Talagrand [16]) Suppose that X is a C -Lipschitz and r -certifiable random variable on Ω = Q ni =1 Ω i as above. Then, P h(cid:12)(cid:12) X − E [ X ] (cid:12)(cid:12) > t + 60 C p r E [ X ] i ≤ e − t C r E [ X ] . Finally we need the symmetric version of the Lov´asz Local Lemma, which is typically used to showthat with positive probability, no “bad” events happen.
Theorem 2.3 (Lov´asz Local Lemma [6]) Let A , . . . , A n be events in a probability space. Suppose thatthere exist constants p and d such that all P [ A i ] ≤ p , and each event A j is mutually independent of allof the other events { A i } except at most d of them. If ep ( d + 1) ≤ , where e is the base of the naturallogarithm, then P (cid:2)T A i (cid:3) > . The following result can be deduced quickly from this lemma. We record it here for later use, andsketch the proof for the sake of completeness.
Proposition 2.4 (Alon [4]) Let G be a multipartite graph with maximum degree ∆ , whose parts V , . . . , V r all have size | V i | ≥ e ∆ . Then G has an independent transversal. The proof follows by applying the Local Lemma to the probability space where we independently anduniformly select one vertex from each V i . For each edge e of G , let the “bad” event A e be when bothendpoints of e are selected. The dependency is bounded by 2(2 e ∆)∆ −
1, and the probability of each A e is at most (2 e ∆) − , so the Local Lemma implies this statement immediately. In this section, we show that it is enough to prove Theorem 1.1 only in the case when the local degreeis bounded by a constant. This will be an immediate consequence of the following claim.
Theorem 3.1
For any ǫ > , there exists γ > such that for all γ < γ and all ∆ , the followingholds. Let G be a multipartite graph with maximum degree ≤ ∆ , parts V , . . . , V r of size | V i | ≥ (1+ ǫ )∆ ,and local degree ≤ γ ∆ . Then there exist subsets W i ⊂ V i , ≤ i ≤ r such that the r -partite subgraph G ′ of G induced by the set S W i has the following properties. The maximum degree of G ′ is at mostsome ∆ ′ > γ − / , each W i has size ≥ (1 + ǫ/ ′ and the local degree of G ′ is less than 10.
4e first prove the following special case of the above theorem, when ∆ / ≤ γ − . Lemma 3.2
For any < ǫ < , there exists ∆ such that the following holds for all ∆ > ∆ . Let G be a multipartite graph with maximum degree ≤ ∆ , parts V , . . . , V r of size | V i | ≥ (1 + ǫ )∆ , andlocal degree ≤ ∆ / . Then there exist subsets W i ⊂ V i , ≤ i ≤ r such that the r -partite subgraph G ′ of G induced by the set S W i has the following properties. The maximum degree of G ′ is at most ∆ ′ = (1 + ǫ/ / , each W i has size at least (1 + ǫ/ ′ and the local degree of G ′ is less than 10. Proof.
By discarding vertices, we may assume that all | V i | = (1 + ǫ )∆. For every 1 ≤ i ≤ r , create W i by choosing each vertex of V i randomly and independently with probability p = ∆ − / . Define thefollowing three types of bad events. For each vertex v , let A v be the event that the number of neighborsof v in W = S W j exceeds (1 + ǫ/ / . For each vertex v and part V i in which v has at least oneneighbor, let B v,i be the event that the number of neighbors of v in W i is at least 10. Finally, forevery 1 ≤ i ≤ r , let C i be the event that | W i | < (1 + 2 ǫ/ / . Note that (1 + 2 ǫ/ / = ǫ/ ǫ/ ∆ ′ ,which exceeds (1 + ǫ/ ′ if ǫ < A v , B v,i is completely determined bythe choices on all vertices within distance one from v , and C k is completely determined by the choiceson all vertices in V k . Since degrees are bounded by O (∆) and all | V k | ≤ O (∆), each event is mutuallyindependent of all but O (∆ ) events.Now we compute the probabilities of bad events. Since the number of neighbors of a vertex v in W is binomially distributed with mean at most ∆ p = ∆ / , the standard Chernoff estimate (Theorem2.1) implies that the probability that it exceeds (1 + ǫ/ / is at most e − Ω(∆ / ) ≪ ∆ − . Similarly,the size of the set W i is binomially distributed with mean at least (1 + ǫ )∆ / . Hence, using theChernoff estimate again, we conclude that P [ C i ] ≤ e − Ω(∆ / ) ≪ ∆ − . Finally, since the number ofneighbors of v in V i is bounded by ∆ / , we have P [ B v,i ] ≤ (cid:18) ∆ / (cid:19) p ≤ ∆ − / ≪ ∆ − Thus, by the Local Lemma, with positive probability none of the events A v , B v,i and C i happen andwe obtain an induced subgraph G ′ of G which has all the desired properties. (cid:3) The general case of Theorem 3.1 cannot be proved using the above arguments, since if γ − weremuch smaller than log ∆, the number of dependencies would overwhelm the probabilities in the ap-plication of the Local Lemma. To overcome this difficulty, we follow an approach similar to the oneused in [5] and construct the desired subgraph by a sequence of random halving steps. This is donevia the following lemma. Lemma 3.3
Let G be a multipartite graph with maximum degree at most ∆ , parts V , . . . , V r each ofsize s , and local degree at most d . Suppose that ∆ is sufficiently large and d > log ∆ . Then thereexist subsets U i ⊂ V i , each of size s , such that the subgraph of G induced by S U i has maximum degreeat most ∆ / / , and local degree at most d/ d / . roof. Within each V i , arbitrarily pair up the vertices so that each vertex v has a mate M ( v ). Notethat this pairing doesn’t need to have any correlation with the original edges of G . For each pair ofvertices { v, M ( v ) } , randomly and independently designate one of the vertices to be in U i . Clearlyall U i will have size s . For each vertex v , let A v be the event that the number of neighbors of v in U = S U i exceeds ∆ / / . Also for every part V k and vertex v V k , let B v,k be the event thatthe number of neighbors of v in U k exceeds d/ d / . We will use the Local Lemma again to provethat with positive probability none of these events occurs.Fix a vertex v and consider the event A v . Note that if two neighbors of v are paired with eachother by M , then exactly one of them will be in U . Let T be the set of all neighbors of v whichare paired by M to vertices which are not neighbors of v . Then the number of neighbors of v in U is at most (∆ − | T | ) / T that belong to U . The second number isbinomially distributed with parameters | T | ≤ ∆ and 1 /
2. Therefore by the Chernoff bound (Theorem2.1), we have that the probability that it deviates from its mean by at least ∆ / is bounded by2 e − / ) / | T | ≪ ∆ − . Using similar arguments, together with the assumption that d > log ∆, wecan bound P [ B v,k ] ≤ e − d / ) /d ≪ ∆ − .To bound the dependency, observe that we can argue exactly as in the proof of the previous lemmato show that every bad event depends on at most O (cid:0) ∆ (cid:1) other such events. Thus, by the Local Lemmawe have that with positive probability none of the events A v , B v,k happen. (cid:3) Proof of Theorem 3.1.
Let G be a multipartite graph with maximum degree ≤ ∆, local degree ≤ d = γ ∆, and parts V , . . . , V r of size (1 + ǫ )∆. First consider the case when γ − / ≥ ∆. Then d ≤ ∆ / , and the result of the theorem follows from Lemma 3.2 because ∆ ′ > ∆ / ≥ γ − / , since∆ ≥ γ − as was noted at the end of the introduction.It remains to consider the case γ − / < ∆. (We choose − / ′ > γ − / .) Let j ≥ j − < γ / ∆ ≤ j . By deleting at most 2 j verticesfrom each V i , we may assume that the size n > (1 + ǫ )∆ − j of every part is divisible by 2 j . Definethe sequences { ∆ t } and { d t } by setting ∆ = ∆, d = d = γ ∆, and∆ t +1 = ∆ t / t , d t +1 = d t d / t . We claim that:( i ) γ − / / < ∆ j ≤ (1 + ǫ/
4) ∆2 j , ( ii ) d j ≤ / j , ( iii ) d t > log ∆ t ∀ ≤ t < j. Suppose this is true. By (iii), we can apply Lemma 3.3 to split each part V i in half and obtain anew r -partite graph G with maximum degree at most ∆ and local degree at most d . Continuingin this manner for j iterations, applying Lemma 3.3 to split the graph in half each time, we obtain asequence of r -partite graphs G ⊃ G · · · ⊃ G j . Note that ∆ t and d t are upper bounds on the maximumand local degrees of each G t , respectively. Moreover, all parts in each G t have size n t = n/ t .By the lower bound of (i), we can make ∆ j as large as necessary by decreasing γ , so the upperbound of (i) yields n j > (1+ ǫ )∆ − j j ≥ ǫ ǫ/ ∆ j − > (1 + ǫ/ j (assume ǫ < d j ≪ ∆ / j .Applying Lemma 3.2 to G j with ǫ/ ǫ , we obtain a new subgraph G ′ with maximum degree6t most ∆ ′ = (1 + ǫ/ / j > ( γ − / / / ≫ γ − / , part sizes at least (1 + ǫ/ ′ , and local degreeless than 10. This completes the proof of the theorem.To finish we need to prove our claim. The lower bound of (i) follows immediately from the definitionof j , because ∆ j ≥ ∆ / j > γ − / /
2. For the upper bound, ∆ t +1 = ∆ t / / t ≤ (cid:0) ∆ / t + 1) , sotaking cubic roots and subtracting 1 / (2 / −
1) from both sides, we obtain∆ / t +1 − / − ≤ / (cid:0) ∆ / t + 1 (cid:1) − / − / (cid:18) ∆ / t − / − (cid:19) . Therefore, ∆ / j − / − ≤ j/ (cid:18) ∆ / − / − (cid:19) , and since ∆ = ∆ and 2 / − > / / j ≤ ∆ / j/ + 4 ≤ (1 + ǫ/ / ∆ / j/ . The last inequality follows from our assumption that γ is small and hence ∆ / j > γ − / / j ≤ (1 + ǫ/ / j . Note that since ∆ j ≥ ∆ t / j − t , we have ∆ t ≤ (1 + ǫ/ / t for all t < j (we will use this in the proof of (iii)).For (ii), the same argument as above (just substitute d t for ∆ t ) shows that d / j ≤ d / j/ + 4 ≤ d / j/ , where the last inequality used that d/ j = γ ∆ / j > γ − / / d j ≤ d/ j = 8 γ ∆ / j .By definition of j and { ∆ t } , γ / ∆ ≤ j , so d j ≤ (cid:0) ∆ / j (cid:1) / ≤ / j .Finally, to prove (iii), note that by definition of j , γ > (∆ / t ) − / for all t < j . Thus d t ≥ d/ t = γ ∆ / t ≥ (∆ / t ) − / (∆ / t ) = (∆ / t ) / ≥ (cid:18) ∆ t ǫ/ (cid:19) / ≫ log ∆ t , and we are done. (cid:3) In this section we obtain the following result, which completes the proof of our main theorem.
Theorem 4.1
For any ǫ > and constant C , the following holds for all sufficiently large ∆ . Let G be a multipartite graph with maximum degree ≤ ∆ , parts V , . . . , V r of size | V i | ≥ (1 + ǫ )∆ , and localdegree ≤ C . Then G has an independent transversal. The proof of this result is based on the approach from [18] together with some additional ideas. Weuse the semi-random method, which constructs an independent transversal in several iterations. Eachiteration is a random procedure, for which we prove that there is a choice of random bits which givedesirable output. We then fix that choice and assume it as the state of affairs for the next iteration.Consider the following random process, which will provide us with one iteration of our algorithm.7. Activate (for this iteration) each remaining part independently with probability 1 / log ∆.2. Uniformly at random select a vertex from each activated part and denote by T the set of allselected vertices.3. For each i and v ∈ V i ∩ T , if v is not adjacent to any w ∈ V j ∩ T with j < i , then add v to theindependent transversal.4. For each vertex v added to the independent transversal in Step 3, delete the entire part containingit from G . Also delete all neighbors of all vertices in T from G .Observe that the deletions ensure that after each iteration, the partial independent transversal con-structed so far has no adjacencies among the remaining vertices. Our objective will be to show thatafter performing several iterations, the remaining graph will have maximum degree ≤ ∆ ′ and parts ofsize ≥ e ∆ ′ , for some ∆ ′ . Then, we will abort the algorithm, and apply Proposition 2.4 to completethe construction of our independent transversal in a single step. In our study of the evolution of degrees and part sizes, the following definitions are useful. For eachpart V i , let s t ( i ) be its size at the start of iteration t . For each vertex v , let N t ( v ) be the set of v ’sneighbors at the start of iteration t , and let d t ( v ) = | N t ( v ) | .Next, define the sequences { S t } and { D t } by setting S = (1 + ǫ )∆, D = ∆, and S t +1 = (cid:18) − ǫ/
4) log ∆ (cid:19) S t , D t +1 = (cid:18) − ǫ/
4) log ∆ (cid:19) D t . Let P ( t ) be the property that at the start of iteration t , all remaining parts have size at least S t , andall remaining vertices v have d t ( v ) ≤ D t . (Completely ignore deleted parts and vertices.) We willprove by induction that there is always a choice of random bits such that we can perform iterationswith property P ( t ) holding for every t ≤ ǫ log ∆. Then at the end of iteration t ′ = ⌈ ǫ log ∆ ⌉ ,all remaining parts have size at least S t ′ +1 and all remaining vertices have degree at most D t ′ +1 . Aroutine calculation reveals that D t ′ +1 S t ′ +1 = (cid:16) − ǫ/
4) log ∆ (cid:17) t ′ D (cid:16) − ǫ/
4) log ∆ (cid:17) t ′ S ≤ − ǫ/
4) log ∆ − ǫ/
4) log ∆ ! t ′ ≤ (cid:18) − ǫ (cid:19)
10 log ∆ ǫ ≤ e − < e . Therefore, by Proposition 2.4 there is an independent transversal through the remaining parts, aspromised above. This will have no adjacencies with the partial independent transversal constructed bythe first t ′ iterations, so the union of the two partial transversals will be a full independent transversal.Note that if t ≤ ǫ log ∆, then S t = Θ(∆) and D t = Θ(∆). We will use this fact throughout therest of the proof. 8t remains to show that if at the beginning of iteration t we have a graph with property P ( t ),then with positive probability the graph obtained at the end of this round satisfies P ( t + 1). Definethe following family of bad events. Let A i be the event that s t +1 ( i ) < S t +1 and let B v be the eventthat d t +1 ( v ) > D t +1 . The dependencies among these events are polynomial in ∆. To see this considerthe auxiliary graph H obtained by adding edges such that every part V i becomes a clique. If weknow the algorithm’s choices on the “patch” consisting of all vertices within distance (with respectto edges in H ) 4 from v , then B v is completely determined. This is because a neighbor w of v canonly be deleted in two ways: either a neighbor of w is selected in Step 2, or the entire part containing w is deleted because a vertex x in that part is selected, but none of x ’s neighbors in lower-indexedparts are selected. So, each event B v is mutually independent from all other events B w whose patchesare disjoint from its own. Since the part sizes are O (∆), the degrees in H are also O (∆), so thedependency is bounded by O (∆ ). Events of type A i are determined by even smaller patches, so thetotal dependency is also O (∆ ). Therefore if we prove that for every part V i and vertex v ( i ) P [ s t +1 ( i ) < S t +1 ] ≪ e − log ∆ log log ∆ and ( ii ) P [ d i +1 ( v ) > D i +1 ] ≪ e − log ∆ log log ∆ , then we can apply the Local Lemma to show that with positive probability none of the events A i , B v occur. This corresponds precisely to property P ( t + 1), completing the induction step. Thus itremains to establish the two probability bounds above. Suppose that our graph has property P ( t ), and let V i be some part of this graph. In this section webound the probability that the size of V i at the end of iteration t is less than S t +1 .For every vertex v and part V k , define d t ( v, k ) to be the number of neighbors of v in part V k at thestart of iteration t . Since D t /S t < D /S = 1 / (1 + ǫ ), by linearity of expectation we have E [ s t +1 ( i )] = X v ∈ V i r Y k =1 (cid:18) − d t ( v, k ) s t ( k ) (cid:19) ≥ X v ∈ V i (cid:18) − P k d t ( v, k ) S t (cid:19) = X v ∈ V i (cid:18) − d t ( v ) S t (cid:19) ≥ s t ( i ) (cid:18) − D t S t (cid:19) > s t ( i ) (cid:18) − ǫ (cid:19) . Instead of proving concentration of s t +1 ( i ) directly, we consider the number of vertices we deletedfrom the part V i in the t -th iteration and prove that this random variable R = s t ( i ) − s t +1 ( i ) isconcentrated. Since the local degree is bounded by C , changing the assignment of any vertex can9hange R by at most C . Also, if R ≥ s , there are at most s vertices in T , each with neighbor(s) in V i ,such that their selection certifies that R ≥ s . Therefore R is C -Lipschitz and 1-certifiable. Note that R ≤ s t ( i ) = Θ(∆) and p E [ R ] ≪ s t ( i ) / log ∆. Thus, using Talagrand’s inequality (Theorem 2.2), weobtain P (cid:20) | R − E [ R ] | > s t ( i )log ∆ (cid:21) < exp (cid:26) − Θ (cid:18) s t ( i )log ∆ (cid:19)(cid:27) ≪ e − log ∆ log log ∆ . Now for sufficiently large ∆, S t +1 ≤ (cid:18) − ǫ/
4) log ∆ (cid:19) s t ( i ) ≤ (cid:18) − ǫ ) log ∆ − ∆ (cid:19) s t ( i ) ≤ E [ s t +1 ( i )] − s t ( i )log ∆ . Note that since we fixed the output of the ( t − s t ( i ) in the definition of R is fixed as well. Thus by linearity of expectation, s t +1 ( i ) − E [ s t +1 ( i )] = E [ R ] − R , so P [ s t +1 ( i ) < S t +1 ] ≤ P (cid:20) s t +1 ( i ) < E [ s t +1 ( i )] − s t ( i )log ∆ (cid:21) ≤ P (cid:20)(cid:12)(cid:12)(cid:12) s t +1 ( i ) − E [ s t +1 ( i )] (cid:12)(cid:12)(cid:12) > s t ( i )log ∆ (cid:21) = P (cid:20)(cid:12)(cid:12) R − E [ R ] (cid:12)(cid:12) > s t ( i )log ∆ (cid:21) ≪ e − log ∆ log log ∆ , which implies (i). In this section we prove that if our graph has property P ( t ) then for every vertex v the probabilitythat its degree at the end of iteration t is greater than D t +1 is ≪ e − log ∆ log log ∆ . Fix a vertex v . If we have d t ( v ) ≤ D t +1 , then we are already done, so suppose that is not the case. Then∆ ≥ d t ( v ) > D t +1 = Θ(∆). For each vertex v , let z t ( v ) be the number of neighbors of v whose entirepart was deleted in Step 4 of iteration t . Clearly d t +1 ( v ) ≤ d t ( v ) − z t ( v ), so if z t ( v ) ≥ d t ( v )log ∆ − Θ (cid:16) d t ( v )log ∆ (cid:17) ,then for sufficiently large ∆ we have d t +1 ( v ) ≤ d t ( v ) − z t ( v ) ≤ (cid:20) − − Θ (cid:18) ∆ (cid:19)(cid:21) d t ( v ) ≤ (cid:20) − ǫ/
4) log ∆ (cid:21) D t = D t +1 . Thus to prove (ii) it is enough to show P (cid:20) z t ( v ) < d t ( v )log ∆ − Θ (cid:18) d t ( v )log ∆ (cid:19)(cid:21) ≪ e − log ∆ log log ∆ . (1)10ecall our notation that for a vertex v and a part V k , d t ( v, k ) is the number of neighbors of v in V k . Call a part V k relevant for v if d t ( v, k ) ≥
1, i.e., v has at least one neighbor in V k . To analyze thebehavior of z t ( v ), we divide the t -th iteration of the algorithm into 2 independent phases. Phase I.
Activate each part relevant for v independently with probability 1 / log ∆, and define therandom variable X = r X k =1 d t ( v, k ) I ( k ) , where the indicator I ( k ) = 1 if part V k was activated and zero otherwise. Randomly select a vertexfrom each of these activated parts, and collect the selected vertices in a set T . Define the subset S ⊆ T as follows. For every i and x ∈ V i ∩ T , we put it in S iff x is not adjacent to any y ∈ V j ∩ T with j < i . Let I ( k ) be an indicator random variable which equals one iff V k ∩ S = ∅ , and define X = r X k =1 d t ( v, k ) I ( k ) . Phase II.
Activate the rest of the parts (i.e., parts that are not relevant for v ) independently, eachwith probability 1 / log ∆, and randomly select a vertex from each of them. Let T be the set of verticesselected in this phase. For each i and u ∈ V i ∩ S , if u is adjacent to some w ∈ V j ∩ T with j < i , thenremove u from S . Define the random variable X = r X k =1 d t ( v, k ) I ( k ) , where the indicator I ( k ) = 1 iff part V k still has at least one vertex in S .Observe that, by definition, the parts relevant for v which we delete entirely during iteration t areexactly the ones with I ( k ) = 1. Therefore z t ( v ) = X ≤ X ≤ X . Our strategy will be to bound z t ( v )by starting from X and working towards X . By linearity of expectation, E [ X ] = P k d t ( v,k )log ∆ = d t ( v )log ∆ .Also, since d t ( v ) = Θ(∆) (see the beginning of this section), local degrees are ≤ C , and the numberof nonzero d t ( v, k ) is at most ∆, we can apply Hoeffding’s inequality (Theorem 2.1) to the sum of theterms in X with d t ( v, k ) = 0 and conclude that P (cid:20) | X − E [ X ] | > d t ( v )log ∆ (cid:21) ≤ ( − C (cid:18) d t ( v )log ∆ (cid:19) ) ≪ e − log ∆ log log ∆ . (2)Next, let us estimate X by studying the difference X − X . Reveal the random selections in theparts activated in Phase I in order of part number (i.e. if i < j and V i and V j were activated, revealthe vertex selection in V i first). For each activated part V i , the difference X − X will gain d t ( v, i )precisely when the selected vertex x ∈ V i ∩ T is adjacent to some selected vertex y ∈ V j ∩ T with j < i . Call such an event a conflict . Its probability is at most CX S t , because there are at most X activated parts with j < i , each of their selected vertices has degree at most C into V i , and | V i | ≥ S t byproperty P ( t ). Now condition on | X − E [ X ] | ≤ d t ( v )log ∆ . If N ≤ X is the number of parts activated11n Phase I, the probability that there are ≥ C d t ( v )log ∆ conflicts is bounded by (cid:18) N C d t ( v )log ∆ (cid:19) (cid:18) CX S t (cid:19) C dt ( v )log2 ∆ ≤ eX C d t ( v )log ∆ CX S t C dt ( v )log2 ∆ ≤ e (cid:0) d t ( v )log ∆ + d t ( v )log ∆ (cid:1) d t ( v )log ∆ S t C dt ( v )log2 ∆ ≤ (cid:18) e + 0 . (cid:19) C dt ( v )log2 ∆ ≪ e − log ∆ log log ∆ . Here we used that S t ≥ d t ( v ) and ∆ is sufficiently large. Since all d t ( v, i ) ≤ C , each conflict can accountfor a value gain of at most C in X − X . Therefore, we proved that conditioned on | X − E [ X ] | ≤ d t ( v )log ∆ , P (cid:20) X − X ≥ C d t ( v )log ∆ (cid:21) ≪ e − log ∆ log log ∆ . (3)To estimate X , we will use Talagrand’s Inequality (Theorem 2.2) to show that the difference X − X is strongly concentrated. This requires a Lipschitz condition, so let us first ensure thatwe have a good Lipschitz constant. Let W be the set of vertices which have at least one neighborin some part relevant for v . Since there are at most D t parts relevant for v , it is easy to see that | W | ≤ D t S t ≤ (1 + ǫ )∆ . For w ∈ W , let B w be the event that at least log ∆ neighbors of w areselected for T in Phase I. The number of neighbors of w in a given part is at most C , so the probabilitythat one of them appears in T is ≤ CS t log ∆ , and this happens independently for distinct parts. Since w has neighbors in at most ∆ parts and S t = Θ(∆), we obtain P [ B w ] ≤ (cid:18) ∆log ∆ (cid:19) (cid:18) CS t log ∆ (cid:19) log ∆ ≤ (cid:18) e ∆log ∆ CS t log ∆ (cid:19) log ∆ ≪ e − . . This implies that P h[ B w i ≤ (1 + ǫ )∆ e − . ≪ e − log ∆ log log ∆ . (4)Combining inequalities (2), (3), and (4), we see that P (cid:20)(cid:26) d t ( v )log ∆ − C d t ( v )log ∆ ≤ X ≤ d t ( v )log ∆ + d t ( v )log ∆ (cid:27) ∩ \ B w (cid:21) = 1 − o ( e − log ∆ log log ∆ ) . (5)Crucially, the high probability event in (5) is entirely determined by Phase I, so all of the choicesin Phase II are still independent of it. Now condition on Phase I (i.e., X and I ( k ) are fixed), and alsoon the event in (5). Perform Phase II. We will show that with high probability the random variable R = X − X is small. Observe that since I ≥ I , and we conditioned on Phase I, E [ R ] = r X k =1 d t ( v, k ) E [ I ( k ) − I ( k )] = X ≤ k ≤ r,I ( k )=1 d t ( v, k ) P [ I ( k ) − I ( k ) = 1] . I ( k ) = 1, the difference I ( k ) − I ( k ) will be 1 precisely when the vertex u ∈ V k ∩ S hasone of its (at most D t ) neighbors w selected in Phase II. For each such neighbor w , the probability ofits selection in Phase II is ≤ / ( S t log ∆), so a simple union bound gives P [ I ( k ) − I ( k ) = 1] ≤ D t S t log ∆ .Therefore E [ R ] ≤ X ≤ k ≤ r,I ( k )=1 d t ( v, k ) D t S t log ∆ = X D t S t log ∆ ≤ X log ∆ ≤ Θ (cid:18) d t ( v )log ∆ (cid:19) , since we conditioned on a range for X . Next we show that R is concentrated. We conditioned on T B w , so changing any choice in Phase II can affect R by at most C log ∆. Therefore, R is Lipschitzwith constant C log ∆. It is also clear that R is 1-certifiable. Since d t ( v ) = Θ(∆) and R ≤ X ≤ ∆,by Talagrand’s Inequality (Theorem 2.2) we have P (cid:20)(cid:12)(cid:12) R − E [ R ] (cid:12)(cid:12) > d t ( v )log ∆ (cid:21) ≤ ( − Θ (cid:18) d t ( v )log ∆ (cid:19) C log ∆) E [ R ] !) ≤ exp (cid:26) − Θ (cid:18) ∆log ∆ (cid:19)(cid:27) ≪ e − log ∆ log log ∆ . In particular, P (cid:20) X − X > Θ (cid:18) d t ( v )log ∆ (cid:19)(cid:21) ≪ e − log ∆ log log ∆ . Therefore, with probability 1 − o (cid:0) e − log ∆ log log ∆ (cid:1) , z t ( v ) = X ≥ X − Θ (cid:16) d t ( v )log ∆ (cid:17) ≥ d t ( v )log ∆ − Θ (cid:16) d t ( v )log ∆ (cid:17) . This establishes (1) and completes the proof. (cid:3)
In this section, we study sufficient conditions for the existence of a K s -free transversal in graphs G with maximum degree ∆. Consider s to be a fixed parameter, and let ∆ grow. We will prove that ifthe local degree is o (∆), then parts of size (1 + o (1)) ∆ s − are sufficient.First, let us show that this bound is asymptotically tight via the following construction. Fix anypositive integer n < ∆+1 s − , and let G be a graph with vertex set V = { , . . . , ∆+ 1 }× { , . . . , n } . Let theparts be defined as V i = { ( i, j ) : 1 ≤ j ≤ n } , and let ( i, j ) and ( i ′ , j ) be adjacent for all 1 ≤ i, i ′ ≤ ∆+1.It is clear that G has maximum degree ∆ and local degree 1. We show by contradiction that this graphhas no K s -free transversal. Indeed, if there is such a transversal T , then for each j , the set of vertices( i, j ) ∈ T forms a clique and hence has cardinality at most s −
1. Yet there are only n possibilitiesfor j , so | T | ≤ n ( s − < ∆ + 1. This is a contradiction, since T must have one vertex in each of the∆ + 1 parts. Therefore, parts of size ∆ s − do not guarantee a K s -free transversal. Proof of Theorem 1.2.
Fix ǫ > s ≥
3. Let G = ( V, E ) be a graph with maximum degreeat most ∆ whose vertex set is partitioned into r parts V = V ∪ . . . ∪ V r of size | V i | ≥ (1 + ǫ ) ∆ s − .13olor the vertices of G with s − v , there must be a color c such that the number of neighbors of v which arecolored c is at most (cid:4) ∆ s − (cid:5) . Hence the minimality of the coloring implies that v has at most that manyneighbors in its own color, or else one could obtain a better coloring by changing the color of v to c .Now delete all edges whose endpoints have different colors, and call the resulting graph G ′ . By theabove argument, the maximum degree in G ′ is at most (cid:4) ∆ s − (cid:5) , so G ′ has an independent transversal T by Theorem 1.1. However, T is an ( s − G , and so must be K s -free. (cid:3) Observe that we did not need the local degree condition until we invoked Theorem 1.1. If we donot have a local degree condition, we can apply Haxell’s result [10] instead, which says that parts ofsize 2∆ guarantee an independent transversal in graphs with maximum degree ∆. This immediatelyimplies:
Proposition 5.1
Let G be a graph with maximum degree at most ∆ whose vertex set is partitionedinto r parts V ( G ) = V ∪ . . . ∪ V r of size | V i | ≥ (cid:4) ∆ s − (cid:5) . Then G has a K s -free transversal. Phrased in terms of the function p (∆ , K s ) defined in the introduction, we have p (∆ , K s ) ≤ (cid:22) ∆ s − (cid:23) , which is at most twice Szab´o and Tardos’s lower bound (which they conjectured to be tight) p (∆ , K s ) ≥ ss − s − . Note that for s = 3, it matches their best upper bound, p (∆ , K ) ≤ ∆, which they obtain as aconsequence of a result on acyclic transversals, i.e., transversals which have no cycles. So, this simpleapproach provides an alternate proof of that upper bound. For s >
3, as far as we know, thisproposition gives the current best upper bound. • We proved that if G is a multipartite graph with maximum degree ∆ and local degree o (∆), thenparts of size (1 + o (1))∆ will guarantee an independent transversal. It is interesting to decideif it is possible to achieve the same result under the weaker condition that the number of edgesbetween any pair of distinct parts is o (∆ ). • Let M = M (∆) be the smallest integer such that if G is a multipartite graph with maximumdegree ∆, local degree 1, and parts of size ∆ + M , then it has an independent transversal. Weshowed that M = o (∆) (in fact, this can be sharpened to ∆ − ǫ using our method) and it remainsan interesting problem to better estimate the function M (∆). In particular, an intriguing openquestion is to determine if M (∆) is bounded by absolute constant. Note that a list coloringconstruction of Bohman and Holzman from [8] implies that M would have to be at least 2,because as mentioned in the introduction, an instance of the list coloring problem correspondsto an independent transversal problem with local degree 1.14 Let G be a graph with maximum degree ∆ whose vertex set is partitioned into r equal parts V ( G ) = V ∪ . . . ∪ V r of size n . How large should n be to ensure that we can partition the entiregraph into a disjoint union of n independent transversals? This question is related to the notionof strong chromatic number , see, e.g., [1, 5, 11]. Alon [5] proved that for a (large) constant c ,parts of size n = c ∆ are enough. Haxell [11] reduced the constant to 3, and recently even to3 − ǫ , where ǫ can be as large as 1 / c , which should be at least 2 because of the construction of Szab´o and Tardos mentionedin the introduction.However, if we impose a local degree restriction on G , our result suggests that one does not needparts of size 2∆. We believe that if G has maximum degree ∆ and local degree o (∆) then partsas small as n = (1 + o (1))∆ will guarantee the existence of n disjoint independent transversals.So far we can only prove the much weaker statement that parts of size at least (2 + o (1))∆ aresufficient. This claim follows immediately from our main result together with an argument ofAharoni, Berger, and Ziv. In [1] (see Theorem 5.3) they implicitly proved that if parts of sizeat least f (∆) imply that every vertex v of G is contained in some independent transversal, thenparts of size at least ∆ + f (∆) guarantee the existence of a partition of G into independenttransversals. Our result certainly implies the former statement with f (∆) = (1 + ǫ )∆. Indeed,for any given vertex v , the local degree is o (∆), so we can delete o (∆) neighbors of v from everypart. Then v becomes isolated from rest of the graph. However, the part sizes are still at least(1 + ǫ − o (1))∆ so by Theorem 1.1 we can find an independent transversal among the parts notcontaining v , and then add v . Acknowledgment.
The authors would like to thank Noga Alon, whose suggestion simplified theiroriginal proofs of Theorem 1.2 and Proposition 5.1, and the referees for careful reading of thismanuscript.
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