Tripartite Version of the Corrádi-Hajnal Theorem
aa r X i v : . [ m a t h . C O ] M a y Tripartite Version of the Corr´adi-Hajnal Theorem
Csaba Magyar and Ryan R. Martin ∗† Abstract
Let G be a tripartite graph with N vertices in each vertex class. Ifeach vertex is adjacent to at least (2 / N vertices in each of the otherclasses, then either G contains a subgraph that consists of N vertex-disjoint triangles or G is a specific graph in which each vertex is adjacentto exactly (2 / N vertices in each of the other classes. A central question in extremal graph theory is the determination of the minimumdensity of edges in a graph G which guarantees a monotone property P . If theproperty is the inclusion of a fixed size subgraph H , the answer is given by theclassic theorems of Tur´an [10] (when H is a complete graph) and Erd˝os andStone [4].However, in the case when a graph G is required to contain a spanningsubgraph H ; that is, H has the same number of vertices as G , an importantparameter is a lower bound on the minimum degree that guarantees H is asubgraph of G . Perhaps the most well-known result of this type is a theorem ofDirac [3] which asserts that every n vertex graph with minimum degree at least n contains a Hamiltonian cycle. Another theorem of this type is the so-calledHajnal-Szemer´edi theorem, with the case k = 3 proven first by Corr´adi andHajnal [2]. Theorem 1.1 (Hajnal-Szemer´edi [5])
Let G be graph on n vertices withminimum degree k − k n . If k divides n , then G has a subgraph that consistsof nk vertex-disjoint cliques of size k . A tripartite graph is said to be balanced if it contains the same number ofvertices in each class. Theorem 1.2 is a tripartite version of the Corr´adi-Hajnalresult. ∗ Carnegie Mellon University, Pittsburgh, PA. Email: [email protected] † This author partially supported by DIMACS via NSF grant CCR 91-19999 heorem 1.2 Let G = ( V , V , V ; E ) be a balanced tripartite graph on N vertices such that each vertex is adjacent to at least (2 / N vertices in eachof the other classes. If N ≥ N for some absolute constant N , then G hasa subgraph consisting of N disjoint triangles or G = Γ ( N/ for N/ an oddinteger. The graph Γ ( N/
3) is defined in Section 1.2. The proof is in two parts.Theorem 2.1 in Chapter 2 states that if the degree condition is relaxed, then allgraphs, except a specific class, have the spanning subgraph of disjoint triangles.We will then show how find the spanning subgraph for that excluded class ofgraphs by proving Theorem 3.1 in Chapter 3. Assume that N is divisible by 3.If not, Section 4 shows that the case where N is not divisible by 3 comes as acorollary. Throughout this paper, we will try to keep much of the notation and definitionsin [6]. The symbol + will sometimes be used to denote the disjoint union of sets. V ( G ) and E ( G ) denote the vertex-set and edge-set of the graph G , respectively.The triple ( A, B ; E ) denotes a bipartite graph G = ( V, E ), where V = A + B and E ⊂ A × B . N ( v ) denotes the set of neighbors of v ∈ V . For U ⊂ V \ { v } , N U ( v ) denotes the set of neighbors of v intersected with U . The degree of v is deg( v ) = | N ( v ) | . The degree of v in U is deg U ( v ) = | N U ( v ) | . If H isa subgraph of G , then we relax notation so that deg H ( v ) = deg V ( H ) ( v ). For U ⊂ V , G | U denotes the graph G induced by the vertices U .The graph K is the complete graph on 3 vertices, the “ triangle .” We sayedges and triangles are disjoint if their common vertex set is empty. A balancedtripartite graph on 3 N vertices is covered with triangles if it contains asubgraph of N disjoint triangles. The tripartite version of the Corr´adi-Hajnalresult is Theorem 1.2. When A and B are subsets of V ( G ), we define e ( A, B ) = |{ ( x, y ) : x ∈ A, y ∈ B, { x, y } ∈ E ( G ) }| . For nonempty A and B , d ( A, B ) = e ( A, B ) | A || B | is the density of the subgraph of edges that contain one endpoint in A and onein B . Definition 1.3
The bipartite graph G = ( A, B, E ) is ǫ -regular if X ⊂ A, Y ⊂ B, | X | > ǫ | A | , | Y | > ǫ | B | imply | d ( X, Y ) − d ( A, B ) | < ǫ , otherwise we say G is ǫ -irregular . We will also need a stronger version.2 efinition 1.4 G = ( A, B, E ) is ( ǫ, δ ) -super-regular if X ⊂ A, Y ⊂ B, | X | > ǫ | A | , | Y | > ǫ | B | imply d ( X, Y ) > δ and deg( a ) > δ | B | , ∀ a ∈ A and deg( b ) > δ | A | , ∀ b ∈ B. One of our main tools will be the Regularity Lemma [9], but more specifically,a corollary known as the Degree Form:
Lemma 1.5 (Degree Form of the Regularity Lemma)
For every positive ǫ there is an M = M ( ǫ ) such that if G = ( V, E ) is any graph and d ∈ [0 , isany real number, then there is a partition of the vertex set V into ℓ + 1 clusters V , V , . . . , V ℓ and there is a subgraph G ′ = ( V, E ′ ) with the following properties: • ℓ ≤ M , • | V | ≤ ǫ | V | , • all clusters V i , i ≥ , are of the same size L ≤ ⌈ ǫ | V |⌉ , • deg G ′ ( v ) > deg G ( v ) − ( d + ǫ ) | V | , ∀ v ∈ V , • G ′ | V i = ∅ ( V i are independent in G ′ ), • all pairs G ′ | V i × V j , ≤ i < j ≤ l , are ǫ -regular, each with density either or exceeding d . The above definition is the traditional statement of the Degree Form. Infact, we can guarantee that each cluster that is not V has that all verticesbelong to the same vertex class. The Degree Form is derived from the originalRegularity Lemma (see [8]) which shows that any partition can be refined sothat it is in the form of the Regularity Lemma. The reduced graph G r , hasa vertex set V , . . . , V ℓ with V i ∼ V j if and only if G ′ | V i × V j is ǫ -regular withdensity exceeding d .We will also make use of the so-called Blow-up Lemma. The graph H can be embedded into graph G if G contains a subgraph isomorphic to H . Lemma 1.6 (Blow-up Lemma [7])
Given a graph R of order r and positiveparameters δ , ∆ , there exists an ǫ > such that the following holds: Let N bean arbitrary positive integer, and let us replace the vertices of R with pairwisedisjoint N -sets V , V , . . . , V r (blowing up). We construct two graphs on thesame vertex-set V = ∪ V i . The graph R ( N ) is obtained by replacing all edges of R with copies of the complete bipartite graph K N,N and a sparser graph G isconstructed by replacing the edges of R with some ( ǫ, δ ) -super-regular pairs. Ifa graph H with maximum degree ∆( H ) ≤ ∆ can be embedded into R ( N ) , thenit can be embedded into G . .2 Further Definitions We will frequently refer to the well-known K¨onig-Hall condition, which statesthat if G = ( A, B ; E ) is a bipartite graph, then there is a matching in G thatinvolves all the vertices of A unless there exists an X ⊆ A such that, | N ( X ) | < | X | . Specifically, we often use the immediate corollary that if | A | = | B | andeach vertex in A has degree at least | B | / B has degree atleast | A | /
2, then G must have a perfect matching.With G a k -partite graph, V ( G ) = V + · · · + V k , each V i being a partition,we refer to each V i as a vertex class . We refer to the graph defined by theRegularity Lemma, denoted G r , as the reduced graph of G . G itself is the real graph . Any triangle in G r or in a similar reduced graph is referred toas a super-triangle . A triangle in G is often called a real triangle to avoidconfusion.The notation a ≪ b means that the constant a is small enough relative to b .This has become standard notation in these kinds of proofs. A set is of size γ -approximately M if its size is (1 ± γ ) M . Let us also define two classes of graphs.The first is Θ m × n . The vertices of Θ m × n are { h i,j : i = 1 , . . . , m ; j = 1 , . . . , n } and h i,j ∼ h i ′ ,j ′ iff i = i ′ and j = j ′ . Note that Θ × contains no triangle. Thesecond graph is the graph Γ k . The vertices are { h i,j : i = 1 , . . . , k ; j = 1 , . . . , k } and the adjacency rules are as follows: h i,j ∼ h i ′ ,j ′ if i = i ′ and j = j ′ and either j or j ′ is in { , . . . , k − } . Also, h i,k − ∼ h i ′ ,k − and h i,k ∼ h i ′ ,k for i = i ′ . Noother edges exist. If k is even, then Γ k can be covered by K k ’s, but it cannot if k is odd.For a graph G , define G ( t ) to be the graph formed by replacing each vertexwith a cluster of t vertices and each edge with the complete bipartite graph K t,t .For ǫ ≥ ≥
0, a graph H is ( ǫ, ∆) -approximately G ( t ) if each vertexof G is replaced with a cluster of size ǫ -approximately t and each non-edge isreplaced by a bipartite graph of density at most ∆. For brevity, we will say agraph is ∆ -approximately G ( t ) if it is (0 , ∆)-approximately G ( t ). Note thatif ∆ < ∆ ′ and ǫ ≪ ∆ ′ − ∆, then (if we are allowed to add or subtract vertices toguarantee that clusters are the same size) a graph that is ( ǫ, ∆)-approximately G ( t ) is also ∆ ′ -approximately G ( t ). Let G be a balanced tripartite graph on 3 M vertices such that each vertex in G isadjacent to at least (3 / M vertices in each of the other classes. Proposition 1.7shows that this graph can be covered with triangles. Proposition 1.7 is usedrepeatedly in Section 3. Proposition 1.7
Let G = ( V , V , V ; E ) be a balanced tripartite graph on M vertices such that each vertex is adjacent to at least (3 / M vertices in each ofthe other classes. Then, we can cover G with M vertex-disjoint triangles. roof. Let H be the graph induced by ( V , V ). Each vertex in H is adjacentto at least (3 / M > (1 / M vertices in each of the other classes. Therefore, H can be covered by M disjoint edges. Each of these edges is adjacent to atleast (cid:0) − × (cid:1) M = M/ V and each vertex in V is adjacent toat least M/ V and the M disjoint edges – giving us our M disjoint triangles. ✷ Proposition 1.8 is quite valuable and is used in both Section 2 and Section 3.
Proposition 1.8
For a ∆ small enough, there exists ǫ > such that if H isa tripartite graph with at least − ǫ ) t vertices in each vertex class and eachvertex is nonadjacent to at most (1 + ǫ ) t vertices in each of the other classes.Furthermore, let H contain no triangles. Then, each vertex class is of size atmost ǫ ) t and H is ( ǫ, ∆) -approximately Θ × ( t ) . Proof.
Let ǫ ≪ δ ≪ δ ′ ≪ ∆. First we bound the sizes of the V i . Choosevertices v and v from V ( G ) \ V i such that they form an edge. These verticescan have no common neighbor, giving that | V i | ≤ ǫ ) t .Now choose w ∈ V . Let N ( w ) ∩ V i be written as A i, , for i = 1 ,
2, such thateach vertex in A i, is adjacent to no vertices in A − i, . Furthermore, define A , to be those vertices in V that are adjacent to less than δt vertices in each of A , and A , . The set A , cannot be of size larger than (1 + ǫ ) t . If it were,then there exists an edge in ( A , , A , ). By the degree condition, if δ is smallenough, this edge must have a common neighbor in V \ A , .For all i ∈ [3], remove vertices (if necessary) from the sets A i, to create A ′ i, so that each vertex in A ′ i, is adjacent to less than δt vertices in each A i ′ , for i ′ = i . By the same arguments given before, | A ′ i, | ≤ (1 + ǫ ) t , for i = 1 , ,
3. Asa result, each vertex in A ′ i, is adjacent to less than δ ′ t vertices in each A ′ i ′ , , for i ′ = i . Let A ′ i, = V i \ A ′ i, for i = 1 , , (cid:0) A ′ i, , A ′ i ′ , (cid:1) is sparse.Let v ∈ A ′ , . If N ( v ) ∩ A ′ , = ∅ , then | A , \ N ( v ) | ≤ δt which implies | A , \ N ( v ) | ≤ δt . As a result, | N ( v ) ∪ A , | , | N ( v ) ∪ A , | ≤ (1+ ǫ + δ ) t , imply-ing (cid:12)(cid:12) N ( v ) ∩ A ′ , (cid:12)(cid:12) , (cid:12)(cid:12) N ( v ) ∩ A ′ , (cid:12)(cid:12) ≤ δ ′ t . Similar results occur for w ∈ A ′ , ∪ A ′ , .Once again, it must be true that each (cid:12)(cid:12) A ′ i, (cid:12)(cid:12) ≤ (1 + ǫ ) t .Note that each set A ′ i,j is of size at least (1 − ǫ ) t because the others are ofsize at most (1 + ǫ ) t . Therefore, vertices can be moved from the sets larger than(1 − ǫ ) t to the smaller sets to create sets A ′′ i,j of size in ((1 − ǫ ) t, (1 + ǫ ) t ) withpairwise density at most ∆. ✷ The Fuzzy Tripartite Theorem
Theorem 2.1 allows us, with an exceptional case, to cover G with triangles, evenif the minimum degree is a bit less than (2 / N . Theorem 2.1
Given ǫ ≪ ∆ ≪ , let G = ( V , V , V ; E ) be a balanced tripartitegraph on N vertices such that each vertex is adjacent to at least (2 / − ǫ ) N vertices in each of the other classes. Then, if N is large enough, either G canbe covered with triangles, or G has three sets of size N/ , each in a differentvertex class, with pairwise density at most ∆ . As usual, there is a sequence of constants: ǫ ≪ ǫ ≪ ǫ ≪ ǫ ≪ α ≪ δ ≪ δ ≪ d ≪ d ≪ ǫ ≪ ∆ ≪ ∆Begin with G = ( V , V , V ; E ), a balanced tripartite graph on 3 N verticeswith each vertex adjacent to at least (2 / − ǫ ) N vertices in each of the otherclasses. Define the extreme case to be the case where G has three sets of size N/ d and ǫ , to partition each of the vertex classesinto ℓ + 4 clusters. Let us define G ′ r to be the reduced graph defined by theLemma. It may be necessary to place clusters into the exceptional sets (the setsof vertices in each vertex class that make up the V in Lemma 1.5) to ensure that ℓ is divisible by 3. It is important to observe that in the proof, the exceptionalsets will increase in size, but will always remain of size O ( ǫ ) N .For i = 1 , ,
3, there exist V i = V (0) i + V (1) i + · · · + V ( ℓ +3) i and (cid:12)(cid:12)(cid:12) V ( j ) i (cid:12)(cid:12)(cid:12) = L ≤⌈ ǫ N ⌉ , ∀ i , ∀ j ≥
1. The reduced graph G ′ r has the condition that every cluster isadjacent to at least (2 / − ǫ )( ℓ + 3) clusters in each of the other vertex classes.Apply Lemma 2.2 repeatedly to G ′ r with M = ℓ + 3 to get a decomposition of G ′ r into ℓ vertex-disjoint triangles. If this is not possible, then Lemma 2.2 andProposition 2.3 imply that G is in the extreme case. Lemma 2.2 (Almost-covering Lemma)
Let ǫ ′ ≪ ∆ ≪ , and let G =( V , V , V ; E ) be a balanced tripartite graph on M vertices so that each vertexis adjacent to at least (2 / − ǫ ′ ) M vertices in each of the other classes. If T isa partial cover by disjoint triangles with |T | < M − , then we can find anotherpartial cover by disjoint triangles, T with |T | > |T | and |T \ T | ≤ , unless G contains three sets of size M/ and pairwise have density less than ∆ . Proposition 2.3
If a reduced graph G r has two sets of size ℓ/ and have densityless than ∆ , then some vertices can be added to the underlying graph induced y those clusters so that it is two sets of size ⌊ N/ ⌋ and have density less than ∆ . Call these super-triangles S (1) , S (2) , . . . , S ( ℓ ). We put the vertices in theremaining clusters into the appropriate leftover set. Let the reduced graphinvolving the clusters of S (1) , S (2) , . . . , S ( ℓ ) be denoted G r . By Proposition 2.4,at most 2 ǫ L ′ vertices can be removed from each cluster to obtain ( ǫ , δ )-super-regular pairs in the vertex-disjoint triangular decomposition of G r . Furthermore,Proposition 2.5 guarantees that any edge in G r must still correspond to an ǫ -regular pair of density at least d . Proposition 2.4
Given ǫ < / , let ( S ′ i , S ′ j ) for { i, j } ∈ (cid:0) [3]2 (cid:1) , be three ǫ -regularpairs with density at least d and | S ′ i | = L ′ for i = 1 , , . Some vertices can beremoved from each S ′ i to create S , S and S that form three pairwise (2 ǫ, d − ǫ ) -super-regular sets of size L ≥ (1 − ǫ ) L ′ . Proposition 2.5
Let | X | = | Y | , X ′ ⊆ X , Y ′ ⊆ Y , | X ′ | = | Y ′ | with | X ′ | >ǫ | X | . If ( X, Y ) is ǫ -regular, then ( X ′ , Y ′ ) is max n(cid:16) | X || X ′ | ǫ (cid:17) , ǫ o -regular. One cluster y is reachable from another, x , if there is a chain of super-triangles, T , . . . , T k ( k ∈ { , } ) with x an endpoint of T , and y an endpointof T k with the added condition that T i +1 and T i +2 ( i = 0 , . . . , k −
1) sharea common edge and T i and T i +1 ( i = 1 , . . . , k −
1) share only one commonvertex. Fix one super-regular super-triangle, S (1). The set of all such trianglesthat connects some cluster to a cluster of S (1) is a structure . We would liketo show that each cluster in G r and V i is reachable from the cluster that is S (1) ∩ V i . If this is not possible, then Lemma 2.6 and Proposition 2.3 implythat G must be in the extreme case. Lemma 2.6 (Reachability Lemma)
In the reduced graph G r , all clusters arereachable from other clusters in the same class, unless some edges can be deletedfrom G r so that the resulting graph obeys the minimum degree condition, but is ∆ -approximately Θ × ( ℓ/ . So, suppose that every cluster is reachable from the appropriate cluster of S (1). Consider some cluster y and the structure that connects it to x . Thisstructure contains clusters from at most 8 of the S ( i ), not including S (1) itself.For any such structure, T , . . . , T k , find 3 real triangles in each of the T i , for i odd. Note that if some T is in more than one structure, then there exist 3real triangles for each time that T occurs in a structure. Do this for all possiblestructures, ensuring that these real triangles are mutually disjoint and colorthese real triangles red. No cluster can possibly contain more than r = 9 ℓ red vertices. Thus, there are still L − r uncolored vertices in each cluster, but L ≥ [1 − O ( ǫ )] Nℓ , which goes to infinity as N → ∞ . Proposition 2.7 gives thatfinding these red triangles is easy. 7 roposition 2.7 Let ( X , X , X ) be a triple with | X i | = L for i = 1 , , andeach pair is ǫ -regular with density d > ǫ . Then, there exist (1 − ǫ ) L disjointreal triangles in the graph induced by ( X , X , X ) . This process of creating red triangles may result in an unequal number ofred vertices in the clusters of some of the S ( i )’s. Let s i denote the maximumnumber of red vertices in any one class of S ( i ). Pick a set of uncolored verticesof size L − s i in each class of S ( i ). Proposition 2.5 gives that the pairs of S ( i )are ( ǫ ′ , δ ′ )-super-regular for some ǫ ′ and δ ′ . Then, apply the Blow-up Lemma(Lemma 1.6) to get L − s i disjoint triangles among the uncolored vertices of S ( i ). Color these triangles blue.Now, place the remaining uncovered vertices into the leftover sets. Applythe Almost-covering Lemma (Lemma 2.2) to the non-red vertices of G . Eachtime this is applied, we may end up destroying at most 15 of the blue trianglesin order to create our larger covering. So, suppose that, at some point, thereare less than (1 − δ ) L + 18 vertices remaining in some S ( i ), then we still applythe Almost-covering Lemma, but this time exclude vertices in the blue trianglesof S ( i ) as well as red vertices. There are at most ǫ ℓ of the S ( i )’s that we mayhave to exclude in this manner.Color green any new triangles formed by using the Almost-covering Lemma(Lemma 2.2). There are at most 9 uncolored vertices that remain after we arefinished. Let x ∈ V be an uncolored vertex. We will show how to insert thisvertex; inserting the other vertices is similar.The cluster containing x has degree at least 2 δ L in at least (2 / − α ) ℓ ofthe clusters in V and V . So, choose some S ( i ) where x is adjacent to at least2 δ L vertices in the V and V clusters of S ( i ). Color x blue. Now look atthe structure that connects S ( i ) to S (1), and call the triangles in this structure T , . . . , T k . Find a triangle between the blue vertices of T k . Color the edgesand vertices of this triangle red. Next take one of the red triangles from T k − ,uncolor its edges and color its vertices blue. Continue in the same manner,adding a red triangle to T ξ and removing one from T ξ − for ξ = k, k − , . . . , S ( j ), except for one extra in V ( S (1)) ∩ V .Apply the same procedure to uncolored vertices in V and V . Now, the samenumber of blue vertices are in each S ( j ), including S (1), which now has 9 moreblue vertices in each class than before inserting the extra vertices. Finally, applythe Blow-up Lemma (the pairs are (2 ǫ , δ )-super-regular) to the blue verticesin each of the S ( j )’s to create vertex-disjoint blue triangles that involve all ofthe blue vertices. So, the red, green and blue triangles are vertex-disjoint andcover all vertices of G . ✷ .3 Proofs of Propositions Proof of Proposition 2.3.
This is immediate from the fact that the densityof any pair of clusters nonadjacent in G r is at most d + 2 ǫ and from the factthat ∆ ≪ ∆. ✷ Proof of Proposition 2.4.
Let T be the subset of S ′ consisting of verticeswith degree at most ( d − ǫ ) L ′ in S ′ . Clearly d ( T, S ′ ) ≤ d − ǫ . But, if | T | > ǫL ′ ,then d ( T, S ′ ) > d − ǫ , a contradiction. So, | T | ≤ ǫL ′ . We then have at least(1 − ǫ ) L ′ vertices in S ′ that have degree at least ( d − ǫ ) L ′ in both S ′ and S ′ .Call that set S and similarly define S and S . Proposition 2.5 gives that thesesets are pairwise 2 ǫ -regular if ǫ < /
4, then the proposition is proven. ✷ A proof of Proposition 2.5 is straightforward and left to the reader.
Proof of Proposition 2.7.
We apply Proposition 2.4 to the triple ( X , X , X )to get a triple ( X ′ , X ′ , X ′ ) such that each pair is (2 ǫ, d − ǫ ) super-regular eachon L ∗ ≥ (1 − ǫ ) L vertices. We then apply the Blow-up Lemma (Lemma 1.6)to ( X ′ , X ′ , X ′ ) getting our L ∗ vertex-disjoint triangles. ✷ . Given the constants ǫ ′ ≪ ∆ ′ ≪ ∆ , let T be as in the statement of the lemma.Denote U , U and U as the portions of V , V , and V , respectively, leftuncovered by T . Let U = U + U + U . We want to show that if | U | > G contains three sets of size M/ . (We always assume that M is divisible by3.) Thus, assume that U contains at least four vertices in each class. We wantto show that there are at least three disjoint edges, one between each class. Let x ∈ U and x ∈ U with x x then it will be possible to exchange thesevertices with the vertices of T that maintains or increases the number of disjointtriangles, uses no other vertices in U and places an edge between U and U .By assumption, both (cid:12)(cid:12) N V \ U ( x ) (cid:12)(cid:12) ≥ (2 / − ǫ ′ ) M and (cid:12)(cid:12) N V \ U ( x ) (cid:12)(cid:12) ≥ (2 / − ǫ ′ ) M . This implies that there are at least (1 / − ǫ ′ ) M triangles, T , in T sothat x is adjacent to both the V and V vertices in T .Let A := { x ∈ V : T ∈ T , V ( T ) = { x, y, z } , x ∼ y, and x ∼ z } A := { y ∈ V : T ∈ T , V ( T ) = { x, y, z } , x ∼ y, and x ∼ z } A := { z ∈ V : T ∈ T , V ( T ) = { x, y, z } , x ∼ y, and x ∼ z } Simply, A is the set of all vertices so that x can be exchanged with such avertex so as to leave the number of triangles in T unchanged. The sets A and A are the vertices in the other classes that correspond to the triangles in T with vertices in A . Clearly, | A | = | A | = | A | ≥ (1 / − ǫ ′ ) M .Consider x . Define B , B and B in a similar manner so that x can beexchanged with each of the vertices of B . We will show that the intersection of9 x x x x xx’ x’y’ y’z’ z’before switch after switch Figure 1: ( A , C ) not sparse A and B is empty. If there is a triangle { x, y, z } ∈ T such that x ∈ A ∩ B ,then x and x can be exchanged in order to obtain a covering with the samenumber of triangles but with an edge in ( U , U ).The pair ( A , B ) is void of edges. If not, then both x and x can beexchanged with the endvertices of that edge. The number of triangles doesnot change, but there will be an edge between U and U . Now let C i = V i ( T ) \ ( A i ∪ B i ), for i = 1 , ,
3. Clearly, | C | = | C | = | C | ≤ (1 / ǫ ′ ) M .But, since no vertex in A can be adjacent to a vertex in U and ( A , B ) isvoid, then | C | ≥ (1 / − ǫ ′ ) M .Since ( A , B ) is void, each vertex in A must be adjacent to at least (1 / − ǫ ′ ) M vertices in C . Therefore, if there exists some vertex x ∈ A adjacentto more than 7 ǫ ′ M vertices in C , then there is a triangle { x ′ , y ′ , z ′ } such that x ∼ y ′ , z ′ . Thus, according to Figure 1, x , x ′ and x could be moved so thatthere exists a T of the same size with an edge in ( U , U ).Similarly, ( B , C ) must be sparse. Therefore, the triple ( A , B , C ) has setsof size 4 ǫ ′ -approximately M/ ǫ ′ M/ | C | < ∆ ′ .The same procedure can be applied to ( U , U ) and then ( U , U ) to create 6edges, e , e ∈ ( U , U ), f , f ∈ ( U , U ) and g , g ∈ ( U , U ) that are disjoint.Let this new partial triangular cover be T . Note that |T \ T | ≤
12 but |T | = |T | .Given the edges e , e , f , f , g , g in U , redefine the “ A ”, “ B ”, and “ C ”sets. Let A i = { x i ∈ V i : { x , x , x } ∈ T and x ∼ g i } i = 2 , B i = { x i ∈ V i : { x , x , x } ∈ T and x ∼ f i } i = 1 , C i = { x i ∈ V i : { x , x , x } ∈ T and x ∼ e i } i = 1 , | A i | , | B i | , | C i | ≥ (1 / − ǫ ′ ) M for all relevant i . This is the case becausethe neighborhood of each edge is of this size and these neighborhoods must beentirely within V ( T ), otherwise T is not maximal.We wish to show that these sets are disjoint. Suppose, without loss ofgenerality, x ∈ B ∩ C so that { x , x , x } ∈ T . Then x and f form a10 x x y y y z z z f g g Figure 2: A triangle in ( B , A , A )triangle and x and e form a triangle – giving that there exists a T of sizelarger than T . As a result, | B ∪ C | , | A ∪ C | , | A ∪ B | ≥ (2 / − ǫ ′ ) M .Further, there can be no triangle in the triple ( B ∪ C , A ∪ C , A ∪ B ). Wewill just show one example; suppose there is a triangle T in ( B , A , A ). Thenthere are { x , x , x } , { y , y , y } , { z , z , z } ∈ T such that x ∼ g , y ∼ g , z ∼ f and T = { z , x , y } . If x = y , then { x , x , x } and { z , z , z } canbe replaced with the triangle formed by x and g , the triangle formed by z and f and T itself. If x = y , then we can replace the x , y and z triangleswith the triangles formed by x and g , by y and g and by z and f as wellas T . See Figure 2. Thus, if there were a triangle in ( B ∪ C , A ∪ C , A ∪ B ),then a T could be found such that |T | > |T | and |T \ T | ≤
15. If there isno triangle in ( B ∪ C , A ∪ C , A ∪ B ), then Proposition 1.8 gives that thissubgraph is ∆ -approximately Θ × ( M/ G contains 3 sets ofsize M/ . ✷ Let us be given constants ǫ ≪ ∆ ′ ≪ ∆ ′′ ≪ ∆ ′′′ ≪ ∆ . In order to prove the lemma, we distinguish two triangles, call them S (1) = { x (1) , x (1) , x (1) , } and S ( ℓ ) = { x ( ℓ ) , x ( ℓ ) , x ( ℓ ) } and suppose x ( ℓ ) is notreachable from x (1). We will show that edges can be deleted from G r so that theminimum degree condition holds and the resulting graph is ∆ -approximatelyΘ × ( ℓ/ / − ǫ ) ℓ clusters in each ofthe other classes. Let A i, := [ N ( x ( ℓ )) \ N ( x (1))] ∩ V i A i, := [ N ( x (1)) \ N ( x ( ℓ ))] ∩ V i i = 2 , A i, := V i \ ( A i, ∪ A i, )Observe that (1 / − ǫ ) ℓ ≤ | A i, | , | A i, | ≤ (1 / ǫ ) ℓ for i = 2 , A , , A , ), then x ( ℓ ) must be reachable from x (1).Thus, it must be that d ( A , , A , ) = 0. Combining the information, it mustbe true that | A i,j | ∈ ((1 / − ǫ ) ℓ, (1 / ǫ ) ℓ ) for i ∈ { , } and j ∈ { , , } .Define the sets A , and A , by first letting A , ∪ A , := n v ∈ V : ∃ i ∈ { , } s.t. deg A i, ( v ) ≥ ′ ℓ o . Suppose v ∈ A , ∪ A , with deg A i, ( v ) ≥ ′ ℓ and deg A i ′ , ( v ) , deg A i ′ , ( v ) ≥ ∆ ′′ ℓ , where { i ′ } = { , } \ { i } . Then there exists an edge in ( A i, , A i ′ , ) thatis adjacent to both x (1) and v . Also, there exists another edge in ( A i, , A i ′ , )that is adjacent to both v and x ( ℓ ). This makes x ( ℓ ) reachable from x (1) bya chain of 4 triangles.Suppose v ∈ A , ∪ A , and v is adjacent to less than ∆ ′′ ℓ vertices in A i ′ , butis adjacent to more than ∆ ′′ ℓ vertices in A i, . In this case, there exists an edgein ( A i, , A i ′ , ) that is adjacent to both x (1) and v . There also exists an edge in( A i, , A i ′ , ) that is adjacent to both v and x ( ℓ ). With the above suppositionsabout the degree of v , x ( ℓ ) is reachable from x (1) by a chain of 4 triangles.Therefore, each vertex either is adjacent to less than ∆ ′′ ℓ vertices in both A , and A , (call these vertices A , ) or is adjacent to less than ∆ ′′ ℓ vertices in both A , and A , (call these vertices A , ). This gives that d ( A , , A i, ) < ∆ ′′′ and d ( A , , A i, ) < ∆ ′′′ for i = 2 , | A , | , | A , | < (1 / ǫ ) ℓ . De-fine A , to be those vertices adjacent to less than 2∆ ′ ℓ vertices in both A , and A , . It must be true that V = A , ∪ A , ∪ A , with all sets beingdisjoint, because the definition of A , ∪ A , gives that all vertices not in thosesets must be in A , . From before, | A , | < (1 / ǫ ) ℓ and d ( A , , A i, ) < ∆ ′′′ for i = 2 ,
3. Summarizing, | A i,j | ∈ ((1 / − O ( ǫ )) ℓ, (1 / O ( ǫ )) ℓ ) for all i and j . Furthermore, d ( A , , A i, ) < ∆ ′′′ and d ( A , , A i, ) < ∆ ′′′ for i = 2 , d ( A , , A i, ) < ∆ ′′′ for i = 2 , A , , A , ) or ( A , , A , ) ispairwise sparse. Note that if one is pairwise sparse, we might as well allow theother to be pairwise sparse, since extra edges only help. If it is not the case,then d ( A , , A , ) ≥ ∆ ′′′ and d ( A , , A , ) ≥ ∆ ′′′ are both not sparse. Thereexists an edge e in ( A , , A , ) that is adjacent to many vertices in A , as wellas an edge f in ( A , , A , ) that is adjacent to many vertices in A , . Thus,there exists a vertex v that is adjacent to both edges. Since f is adjacent toboth x (1) and v and e is adjacent to both v and x ( ℓ ) we have that x ( ℓ ) isreachable from x (1). Therefore, if x ( ℓ ) is not reachable from x (1), we musthave that d ( A , , A , ) , d ( A , , A , ) ≪ ∆ . ✷ The Extreme Tripartite Theorem
Theorem 2.1 leaves the extreme case, which we consider in Theorem 3.1.
Theorem 3.1
Given ∆ ≪ , let G = ( V , V , V ; E ) be a balanced tripartitegraph on N vertices such that each vertex is adjacent to at least (2 / N verticesin each of the other classes. Furthermore, let G have three sets with size N/ and pairwise density at most ∆ . Then, if N is large enough, either G can becovered with triangles or G is Γ ( N/ . Assume that G is minimal. That is, no edge of G can be deleted so that theminimum degree condition still holds. We will prove that for minimal G , either G can be covered with triangles or G = Γ ( N/ ( N/
3) will allow the resultant graph to becovered with triangles – this will be discussed in Section 3.3. Begin with theusual sequence of constants:∆ ≪ ∆ ≪ ∆ ≪ η ≪ θ − θ , 3 / < θ <
1. Let t := N/ N divisible by 3.Let the sets of size t mentioned in the theorem be designated A i , with A i ⊂ V i for i = 1 , ,
3. Let B i := V i \ A i for i = 1 , ,
3. For each i ∈ { , , } , let A ′ i bethe vertices that are adjacent to at least (1 + θ ) t vertices in B j for each j = i .Let B ′ i be the vertices that are adjacent to at least (1 / θ ) t vertices in A j for each j = i . Furthermore, let C ′ i = V i \ ( A ′ i ∪ B ′ i ). The key feature of each c ∈ C ′ i is that there is a j = i such that c is adjacent to at least (1 − θ ) t verticesin A j . Let us compute | A ′ i | and | B ′ i | for i = 1 , ,
3. Proposition 3.2 restricts thesizes of these sets.
Proposition 3.2 If ∆ ≪ ∆ , then for all i ∈ { , , } , | A ′ i | ∈ ((1 − ∆ ) t, (1 + ∆ ) t ) | B ′ i | ∈ ((2 − ∆ ) t, (2 + ∆ ) t ) Furthermore, for i = 1 , , , | A i \ A ′ i | , | B i \ B ′ i | ≤ ∆ t . The key lemma for this proof is Lemma 3.3.
Lemma 3.3
Let ∆ ≪ ∆ ′ ≪ ∆ ′′ ≪ ∆ ′′′ ≪ ∆ ′ ≪ ∆ ≪ θ − for someconstant θ > / . Let G = ( V , V , V ; E ) be a balanced tripartite graph on t ′ vertices with each vertex adjacent to at least (2 − ∆ ′ ) t ′ vertices in each ,12,11,1 1,22,23,2 3,32,31,3 hhh hhh hhh Figure 3: Diagram of Γ . The dotted lines correspond to non-edges. of the other vertex classes. Suppose further that we have sets A ′ i of size ∆ ′′ -approximately t ′ such that for all a ∈ A ′ i , deg V j \ A ′ j ( a ) ≥ (1 + θ ) t ′ for all j = i .Furthermore, let d ( A ′ i , A ′ j ) < ∆ ′′′ , ∀{ i, j } ∈ (cid:0) [3]2 (cid:1) and let each v ∈ V i \ A ′ i havethe property that there is a j = i such that deg A ′ j ( v ) ≥ (1 − θ − ∆ ′′ ) t ′ . If G isminimal and cannot be covered with triangles then either1. | A ′ | + | A ′ | + | A ′ | > t ,2. G is ∆ ′ -approximately Γ ( t ′ ) , or3. G is ∆ ′ -approximately Θ × ( t ′ ) . We make adjustments according to whether or not | A ′ | + | A ′ | + | A ′ | ≤ t . Itis true that, 3(1 − ∆ ) t ≤ | A ′ | + | A ′ | + | A ′ | ≤ ) t . If | A ′ | + | A ′ | + | A ′ | ≤ t ,then apply Lemma 3.3 to G . Thus, G can be covered with triangles unless G is∆ -approximately Γ ( t ) or G is ∆ -approximately Θ × ( t ).If | A ′ | + | A ′ | + | A ′ | > t , then we want to create a matching of size | A ′ | + | A ′ | + | A ′ | − t in ( A ′ , A ′ , A ′ ). After finding the matching, find commonneighbors in ( B ′ ∪ C ′ , B ′ ∪ C ′ , B ′ ∪ C ′ ) and remove those disjoint triangles sothat Lemma 3.3 can be applied to the remaining graph. Each vertex in A ′ i isadjacent to at least max {| A ′ j |− t, } vertices in A ′ j , for all distinct i and j . Thus,we can create matchings sequentially in ( A ′ i , A ′ j ), for all pairs ( i, j ) so that theydo not coincide and together they exclude exactly t vertices in each of A ′ , A ′ and A ′ that are larger than t . The details are left to the reader.Thus, G can be covered with triangles unless G is either ∆ -approximatelyΓ ( t ) (Section 3.3) or ∆ -approximately Θ × ( t ) (Section 3.4). G is ∆ -approximately Γ ( t ) (Case (2) of Lemma 3.3) Let the sets A i,j , i, j = 1 , , h i,j in Figure 3. Note that the figuredepicts the non-edges of this graph. Each row of vertices corresponds to a vertexclass and the dotted lines correspond to non-edges. Given ∆ ≪ ∆ ≪ ∆ ≪ η ,14ur goal is to modify the sets A i,j to form sets ˜ A i,j . The triangles will comefrom each of the following:( ˜ A , , ˜ A , , ˜ A , ) ( ˜ A , , ˜ A , , ˜ A , ) ( ˜ A , , ˜ A , , ˜ A , )( ˜ A , , ˜ A , , ˜ A , ) ( ˜ A , , ˜ A , , ˜ A , ) ( ˜ A , , ˜ A , , ˜ A , ) . The triangles will receive one of 6 labels ( i ; j ), for i ∈ { , , } and j ∈ { , } . Atriangle with the label ( i ; j ) will be in the triple ( ˜ A i, , ˜ A i ,j , ˜ A i ,j ), where i , i are distinct indices in { , , } \ { i } .Define A ′ i,j to be the set of “typical” vertices in A i,j . That is, if { h i ,j , h i ,j } is a non-edge in Γ , then each vertex in A ′ i ,j is adjacent to less than ηt verticesin A i ,j . Let C i = V i \ (cid:0) A ′ i, ∪ A ′ i, ∪ A ′ i, (cid:1) , for i = 1 , ,
3. Since ∆ ≪ ∆ , | A i,j \ A ′ i,j | < ∆ t . We will make the sets A ′ i, into sets A ′′ i, of size t , for i = 1 , ,
3. If there is some | A ′ i, | > t , then we find a matching in ( A ′ , , A ′ , , A ′ , )of size P i =1 max {| A ′ i, | − t, } similar to the one we constructed above. Colorthis matching red and for | A ′ i, | > t , take | A ′ i, | − t red edges and remove the V i endvertices that are in A ′ i, and add them to one of A ′ i, or A ′ i, , whichever hassize smaller than t . This creates sets A ′′ i, of size at most t , for i = 1 , ,
3. Theendvertices of this red edge will receive label ( i ; j ) if one of its vertices is addedto A ′ i,j .Suppose that | A ′ i, | < t . Then find vertices in either A ′ i, or A ′ i, , color themgreen and add them to A ′ i, to form A ′′ i, . To show that these green verticeswill act as A ′ i, vertices, suppose, without loss of generality, v is a green vertexadded to A ′ , . Observe that v must be adjacent to at least (1 − η ) t vertices ineither A , and A , or A , and A , . Thus, if we move a green vertex from A ′ i,j , it will receive the label ( i ; j ). The resulting sets A ′′ i, are of size exactly t ,so let them be renamed ˜ A i, , i = 1 , , C ′ i behave like vertices in either A ′ i, or A ′ i, . Let c ∈ C ′ and, without loss of generality, show that c can be addedto A ′ , . There exists an i ∈ { , } such that c is adjacent to at least ηt verticesin A i, . If c is adjacent to at least ηt vertices in A − i, , then c can receive thelabel ( i ; 2). Otherwise c can receive the label (5 − i ; 2). Color the C ′ i verticesgreen and add them to either A ′ i, or A ′ i, (the smaller of the two) to form A ′′ i, and A ′′ i, .Unfortunately, one of the sets A ′′ i, or A ′′ i, might be of size more than t . Inorder to create sets of size t , let us suppose without loss of generality, that both | A ′′ , | > t and A ′′ , is the largest from among A ′′ , , A ′′ , , A ′′ , and A ′′ , . Let τ = | A ′′ , | − t and observe that A ′′ , = A ′ , . Let q = | A ′′ , | − | A ′′ , | ≤ τ because | A ′′ , | ≥ | A ′′ , | . Let W ⊂ A ′′ , ∩ A ′ , , (cid:0) | A ′′ , | − t (cid:1) | W | ≤ e ( W, A ′′ , ) ≤ ( γ + 2∆ ) t (cid:12)(cid:12)(cid:12) N A ′′ , ( W ) (cid:12)(cid:12)(cid:12) . So, | N A ′′ , ( W ) | ≫ τ , provided | W | is not too small, and there exists a matchingof size q in ( A ′′ , , A ′′ , ∩ A ′ , ). Color this matching blue.15f | A ′′ , | > t , take | A ′′ , | − | A ′′ , | blue vertices from A ′′ , and add them to A ′′ , . Also, take | A ′′ , | − t edges in ( A ′′ , , A ′′ , ), color them blue and add theirvertices to A ′′ , and A ′′ , . Such blue edges will be in triangles with label (3; 3).If | A ′′ , | ≤ t , then take t − | A ′′ , | blue vertices from A ′′ , and add them to A ′′ , .The endvertices of these blue edges will be in triangles labeled (3; 2). For theremaining blue edges, take (cid:12)(cid:12) A ′′ , (cid:12)(cid:12) − t of the vertices from A ′′ , and add them to A ′′ , . The endvertices of these blue edges will be in triangles labeled (3; 3).It may be necessary to find a similar matching in ( V , V ) if either | A ′′ , | > t or | A ′′ , | > t . It is easy to see that we can do so without using any of the othercolored vertices by choosing a W that excludes blue vertices. The sets thatresult from moving the vertices of blue edges are of size exactly t , so denotethem ˜ A i,j , for i = 1 , , j = 2 , ( t ) cannot be covered with triangles if t is odd. A similardilemma must also be resolved in this case. Suppose t is odd. Our goal is tofind three triangles in G such that each vertex is from a different ˜ A i,j . Callthese parity triangles . To find them, we look for an edge in ( ˜ A , ∪ ˜ A , ∪ ˜ A , , ˜ A , ∪ ˜ A , ∪ ˜ A , ) with a common neighbor in ˜ A , ∪ ˜ A , ∪ ˜ A , .If there is such an edge among uncolored vertices, then there are manyneighbors in ˜ A , ∪ ˜ A , ∪ ˜ A , . If any vertex was colored, then by re-examiningthe process by which it was constructed we see that it is possible to createsuch a triangle. For example, if there is a red edge in ( ˜ A , , ˜ A , ), then we canfind a common neighbor in ˜ A , . In any case, if the desired triangle is found,remove it, along with two other triangles so that the resulting “ ˜ A ” sets are ofsize t −
1. If a parity triangle cannot be created in this way, then G contains nocolored vertices. In that case, if there is an edge in the graph that is inducedby ˜ A , ∪ ˜ A , ∪ ˜ A , , then it is easy to create the parity triangles. Otherwise, G = Γ ( t ) and the theorem would be proven.Therefore, suppose that the remaining ˜ A i,j sets are of the same even cardi-nality. Partition each ˜ A i,j uniformly at random into two equally-sized pieces.Each piece will receive one of six labels ( i ; j ), for i = 1 , , j = 2 ,
3. For i ∈ { , , } , ˜ A i, will be partitioned into one set labeled ( i ; 2) and the other la-beled ( i ; 3). For i ∈ { , , } and j ∈ { , } , ˜ A i,j will be partitioned into one setlabeled ( i ; j ) and the other labeled ( i ; j ), where i, i , i are distinct membersof { , , } .The triangle cover will only consist of triangles with vertices in pieces withthe same label. Each of the colored vertices corresponds to at least one of thetwo labels, but not necessarily both. For example, if there is a red edge in( ˜ A , , ˜ A , ), then we want to ensure that each of its endvertices are in pieceslabeled (2; 2). So, it may be necessary to exchange colored vertices in one piecewith uncolored ones in the other piece. A total of at most ∆ t vertices will beso exchanged in any ˜ A i,j .The covering by triangles can be completed by taking each piece that hasthe same label and covering the corresponding triple with triangles. Consider,16 ,1 1,2 1,32,32,22,13,1 3,2 3,3 hhh hhh hhh Figure 4: Diagram of Θ × . The dotted lines correspond to non-edges.for example, the vertices in ˜ A , , ˜ A , and ˜ A , that carry the label (1; 2). Forsimplicity, call them S , S and S , respectively.Any green vertex v ∈ S is adjacent to at least ηt vertices in both A , and A , . Thus, it is adjacent to at least ( η − ) t vertices in both ˜ A , and ˜ A , .Since the S i were chosen at random, Stirling’s inequality (see Corollary 3.5 inSection 3.7) gives that v is adjacent to at least ( η − O (∆ ))( t/
2) uncolored andunexchanged vertices in both S and S . Since all but O (∆ ) t of the verticesin A , have degree at least (1 − ) t in A , , Stirling’s inequality again givesthat there exists an edge among the uncolored and unexchanged vertices of( N ( v ) ∩ S , N ( v ) ∩ S ). Do this for all the green vertices in order to get disjointgreen triangles.The red and blue edges are even easier. For example, since each endvertexof a red edge in ( S , S ) is adjacent to at least (1 − η ) t vertices in A , , we canfind a common vertex among the uncolored and unexchanged vertices of S .So, extend the colored edges to find red and blue triangles disjoint from eachother and from the green triangles. Finally each uncolored vertex in S i that was“exchanged” has degree at least (1 − η − O (∆ ))( t/
2) in each of the S j , ∀ j = i .Put these in black triangles disjoint from each other and from other coloredtriangles. Let there be t ′ ≥ (1 − O (∆ ))( t/
2) uncolored vertices remaining ineach class. Call them S ′ i ⊂ S i , for i = 1 , ,
3. Since ∆ ≪ η , each vertex in S ′ i is adjacent to at least (3 / t ′ vertices in each of the S j , ∀ j = i . Proposition 1.7finishes the covering and the proof of this case. G is ∆ -approximately Θ × ( N/ (Case (3) of Lemma 3.3) Let the sets A i,j , i, j = 1 , , h i,j in Figure 4. Note that the figuredepicts the non-edges of this graph. Each row of vertices corresponds to a vertexclass and the dotted lines correspond to non-edges. Our goal is again to modifythe sets A i,j to form sets ˜ A i,j . Our triangles will come from (cid:16) ˜ A i , , ˜ A i , , ˜ A i , (cid:17) for distinct i , i , i . Triangles that come from this triple will receive the label( i , i ). 17he method is very similar to that in Section 3.3. The sets A ′ i,j , A ′′ i,j , ˜ A i,j and C ′ i will be created similarly to before. But this case is easier not only be-cause each c ∈ C i can be added to any set A ′ i,j for any j ∈ { , , } and it is easyto find the parity triangles. Parity triangles can be found in uncolored verticesof (cid:16) ˜ A , , ˜ A , , ˜ A , (cid:17) , (cid:16) ˜ A , , ˜ A , , ˜ A , (cid:17) and (cid:16) ˜ A , , ˜ A , , ˜ A , (cid:17) . Partitioning the˜ A i,j sets in half uniformly at random, exchanging the colored vertices and ap-plying Proposition 1.7 finishes the proof. ✷ Let X be the set of vertices in A that are adjacent to less than (1 + θ ) t verticesin B . Computing the densities,∆ | A || A | ≥ e ( A , A ) ≥ t | A | − | A || B | + | X | [ | B | − (1 + θ ) t ] . So, it must be true that | X | ≤ | A | | B | − t + ∆ | A || B | − (1 + θ ) t ≤ ∆1 − θ t. As a result, | A i \ A ′ i | ≤ − θ t . Similarly, | B i \ B ′ i | ≤ − θ t . With ∆ ≥ − θ theproposition is proven. ✷ Again, there are a sequence of constants:∆ ′′′ ≪ δ ≪ δ ≪ δ ≪ δ ≪ δ ≪ δ ≪ δ ≪ δ ≪ ∆ ′ . Begin by defining B ′ i = n v ∈ V i : deg A ′ j ( v ) ≥ (1 / θ ) t ′ , ∀ j = i o , for i = 1 , , C ′ i = V i \ ( A ′ i ∪ B ′ i ). Again, using Proposition 3.2, we see that | C ′ i | ≤ δ t ′ .Now, we find 3 t ′ −| A ′ |−| A ′ |−| A ′ | disjoint triangles in ( B ′ ∪ C ′ , B ′ ∪ C ′ , B ′ ∪ C ′ ).If this is not possible, Proposition 1.8 gives that G must be ∆ ′ -approximatelyΘ × ( t ′ ).If such disjoint triangles exist, then remove them from the graph to create B ′′ i and C ′′ i for i = 1 , ,
3. All that remains to prove is that there exists a matching, M , in ( B ′′ ∪ C ′′ , B ′′ ∪ C ′′ , B ′′ ∪ C ′′ ) such that for any triple { i , i , i } , there isa matching in ( B ′′ i ∪ C ′′ i , B ′′ i ∪ C ′′ i ) of size | A ′ i | with each c ∈ C ′′ i is adjacentto at least (1 − θ − δ ) t ′ vertices in A ′ i . What we will do is first form trianglesthat involve the c vertices and then, because δ ≪ θ − /
4, we can see thateach remaining vertex in, say A ′ , is adjacent to at least half of the edges in theportion of the matching that is in ( B ′′ ∪ C ′′ , B ′′ ∪ C ′′ ) and each edge of this18ortion of the matching is adjacent to at least half of the remaining vertices in A ′ . K¨onig-Hall gives that there must be a covering by triangles.In order to find this matching, we will randomly partition the sets B ′′ i ∪ C ′′ i .Let B ′′ i ∪ C ′′ i = S i ( j ) ∪ S i ( k ), where { j, k } = { , , } \ { i } and | S i ( j ) | = | A ′ j | forall distinct i and j . It is important to take note that with probability 1 − o (1),and for all vertices v in the graph,deg S i ( j ) ( v ) − (cid:18) | A ′ j || B ′′ i ∪ C ′′ i | (cid:19) deg B ′′ i ∪ C ′′ i ( v ) ∈ ( − o ( t ′ ) , + o ( t ′ )) . This is a result of Stirling’s inequality [1] (see Section 3.7).Once the “ S ” sets are randomly chosen, it may be necessary to move the“ C ′′ ” vertices. Let us suppose that c ∈ C ′′ i ∩ S i ( j ) is not adjacent to atleast (1 − θ − δ ) t ′ vertices in A ′ j . Then, we will exchange c with a vertexin B ′′ i ∩ S i ( k ) (where k = { , , } \ { i, j } ). Do this for all i and all c ∈ C ′′ i andthen match each moved vertex in S i ( j ) with an arbitrary neighbor in S k ( j ) (for j = k ). Color these edges red. There are at most δ t ′ red edges in any pair( S i ( j ) , S k ( j )). Then, finish by finding a matching between the uncolored verticesof ( S i ( j ) , S k ( j )). If this is not possible, then Proposition 3.4, a simple conse-quence of K¨onig-Hall, gives that edges can be removed so that the minimumdegree condition holds, but the pairs must be δ -approximately Θ × ( | A ′ j | / Proposition 3.4
Let ǫ ≪ ∆ and G = ( V , V ; E ) be a balanced bipartite graphon M vertices such that each vertex is adjacent to at least (cid:0) − ǫ (cid:1) M verticesin the other class. If G has no perfect matching, then some edges can be deletedso that the minimum degree condition is maintained and G is ∆ -approximately Θ × ( M/ . If, with probability at least 2 /
3, the pair ( S i ( j ) , S k ( j )) has such a match-ing, then we complete the triangle cover via K¨onig-Hall and the proof is com-plete. Otherwise, with probability at least 1 /
3, the pair ( S i ( j ) , S k ( j )) is δ -approximately Θ × ( | A ′ j | / δ ≪ δ ≪ δ , ( B ′′ i ∪ C ′′ i , B ′′ k ∪ C ′′ k ) itself is ( δ , δ )-approximately Θ × ( t ′ ).We want to show that, unless all three pairs are ( δ , δ )-approximatelyΘ × ( t ′ ), the matching M exists. Without loss of generality, suppose that( B ′′ ∪ C ′′ , B ′′ ∪ C ′′ ) is not ( δ , δ )-approximately Θ × ( t ′ ). Then choose “ S ” setsas before and move the “ C ” vertices as before. A matching exists among the un-colored vertices of ( S (3) , S (3)) that involves all but O ( δ ) t ′ vertices. But thenexchange vertices – outside of this matching – in S (3) with vertices in S (1)so that M can be completed. If necessary, do the same with S (2) and S (1).Color the edges formed by the switching red. If there does not exist a matchingamong the uncolored vertices in ( S (1) , S (1)), then, as before, we must havethat ( B ′′ ∪ C ′′ , B ′′ ∪ C ′′ ) is ( δ , δ )-approximately Θ × ( t ′ ), a contradiction. So,each pair ( B ′′ i ∪ C ′′ i , B ′′ k ∪ C ′′ k ) must be ( δ , δ )-approximately Θ × ( t ′ ).19he objective is to show that the subsets of vertices involved in forming the( δ , δ )-approximately Θ × ( t ′ ) must have a trivial intersection. Write ( B ′′ i ∪ C ′′ i , B ′′ j ∪ C ′′ j ) as ( P i → j ( a ) ∪ P i → j ( b ) , ( P j → i ( a ) ∪ P j → i ( b )) where each of the “ P ”sets are of size δ -approximately t ′ and d ( P i → j ( a ) , P j → i ( b )) , d ( P i → j ( b ) , P j → i ( a ))) < δ . Suppose, without loss of generality, that | P → ( a ) ∩ P → ( a ) | , | P → ( a ) ∩ P → ( b ) | , | P → ( b ) ∩ P → ( a ) | , | P → ( b ) ∩ P → ( b ) | ≥ δ t ′ . Then, as in the paragraph above, we can simply choose “ S ” sets of appropriatesize at random. Exchange vertices so as to force a matching in ( S (3) , S (3)) andthen, since the intersections of the “ P ” sets are so large, it is easy to exchangevertices in B ′′ ∪ C ′′ so that matchings are forced in both ( S (2) , S (2)) and( S (1) , S (1)). Using K¨onig-Hall to complete the covering by triangles gives usa contradiction.Since, within a tolerance of δ t ′ , the “ P ” sets coincide, we may assumethat P → ( a ) and P → ( a ) coincide and that P → ( a ) and P → ( a ) coincide.Therefore, the issue is whether P → ( a ) and P → ( a ) coincide or whether theyare virtually disjoint. If they coincide, then G is ∆ ′ -approximately Γ ( t ′ ). Ifthey are disjoint, then G is ∆ ′ -approximately Θ × ( t ′ ). ✷ Stirling’s inequality (see, for example, [1]) is a well-known result that gives b ( n, k ) exp (cid:20) − (cid:18) n − k + 1 k (cid:19)(cid:21) ≤ (cid:18) nk (cid:19) ≤ b ( n, k ) exp (cid:20) (cid:18) n (cid:19)(cid:21) . if b ( n, k ) = n n ( n − k ) ( n − k ) k k q n πk ( n − k ) . The proofs of Theorem 3.1 will use thefollowing corollary: Corollary 3.5 If G is a graph on n vertices and X is a set on Ω( n ) verticesthen, with ǫ ≪ p and n large enough, if X ′ is chosen uniformly from (cid:0) Xp | X | (cid:1) , Pr {| deg X ′ ( v ) − p deg X ( v ) | ≤ ǫn, ∀ v ∈ V ( G ) \ X } → as n → ∞ . N is Not a Multiple of 3 We have proven the theorem for the case where N/ t be an integer so that N = 3 t + 1 and let N = 3 t be large enoughso that Theorem 1.2 is true for all multiples of 3 larger than N . Remove anytriangle from G to form the graph G ′ . Then, since every vertex in G is adjacentto at least ⌈ t + 2 / ⌉ = 2 t + 1 vertices in each of the other classes of G , everyvertex in G ′ is adjacent to at least 2 t vertices in the other classes of G ′ . If G ′ can be covered with triangles, then clearly G can also. If G ′ = Γ ( t ), for t odd, then each vertex in G ′ must be adjacent to both the vertices in the othervertex classes of G \ G ′ . Therefore, by removing a triangle in ( A , , A , , A , )(with vertex clusters of G ′ labeled similarly to the diagram in Figure 4) andremoving triangles formed by the vertices of V ( G ) \ V ( G ′ ) and edges that spanthe remaining A sets, the resulting graph, G ′′ is Γ ( t − N = 3 t +2 is similar. Remove 2 disjoint triangles. The resultinggraph can either be covered by disjoint triangles or it is Γ ( t ), for t odd. In thatcase we do the same as in the previous paragraph, forming 5 disjoint trianglesand what remains is the graph Γ ( t − An interesting question is that of how to eliminate the phrase “if N is largeenough.” It may not be necessary to write another proof. In fact, Conjecture 5.1would give us a proof of Theorem 1.2 with N = 1. Conjecture 5.1
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