On deficiency problems for graphs
aa r X i v : . [ m a t h . C O ] F e b ON DEFICIENCY PROBLEMS FOR GRAPHS
ANDREA FRESCHI, JOSEPH HYDE AND ANDREW TREGLOWN
Abstract.
Motivated by analogous questions in the setting of Steiner triple systems and Latinsquares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combi-natorial Theory Series B, 2020] recently introduced the notion of graph deficiency . Given a globalspanning property P and a graph G , the deficiency def( G ) of the graph G with respect to theproperty P is the smallest non-negative integer t such that the join G ∗ K t has property P . Inparticular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an n -vertex graph G needs to ensure G ∗ K t contains a K r -factor (for any fixed r ≥ G ∗ K t to contain anyfixed bipartite ( n + t )-vertex graph of bounded degree and small bandwidth. Introduction
A natural question dating back to the 1970s asks for the order of the smallest complete Steinertriple system a fixed partial Steiner triple system can be embedded into (see e.g. [3, 10, 11]).Similarly, there has been interest in establishing the order of the smallest Latin square that a fixedpartial Latin square can be embedded into (see e.g. [5, 6, 11]).Motivated by these research directions, Nenadov, Sudakov and Wagner [11] introduced the notionof graph deficiency : for a graph G and integer t ≥
0, denote by G ∗ K t the join of G and K t , whichis the graph obtained from G by adding t new vertices and adding all edges incident to at leastone of the new vertices. Given a global spanning property P and a graph G , the deficiency def( G )of the graph G with respect to the property P is the smallest t ≥ G ∗ K t hasproperty P .One of the main results in [11] is a bound on def( G ) with respect to the Hamiltonicity propertyfor graphs G of a given density. More precisely, the following result answers the question of howmany edges an n -vertex graph G can have such that G ∗ K t does not contain a Hamilton cycle. Theorem 1.1 (Nenadov, Sudakov and Wagner [11]) . Let n and t be integers and G an n -vertexgraph so that G ∗ K t does not contain a Hamilton cycle. Then we have the following bounds on e ( G ) . • If n + t is even: e ( G ) ≤ (cid:18) n (cid:19) − (cid:0) t ( n − − (cid:0) t (cid:1)(cid:1) if t ≤ ( n + 4) / (cid:16)(cid:0) n + t +22 (cid:1) − (cid:17) if t ≥ ( n + 4) / . • If n + t is odd: e ( G ) ≤ (cid:18) n (cid:19) − (cid:0) t ( n − − (cid:0) t (cid:1)(cid:1) if t ≤ ( n + 1) / (cid:0) n + t +12 (cid:1) if t ≥ ( n + 1) / .These bounds on e ( G ) are sharp. Another line of inquiry in [11] concerns the deficiency problem for K r -factors. Given graphs H and G , an H -factor in G is a collection of vertex-disjoint copies of H in G that together cover all thevertices of G . Note that H -factors are also often referred to as perfect H -tilings , perfect H -packings r perfect H -matchings . The following seminal result of Hajnal and Szemer´edi [7] determines theminimum degree threshold for forcing a K r -factor in a graph G . Theorem 1.2 (Hajnal and Szemer´edi [7]) . Every graph G on n vertices with r | n and whose mini-mum degree satisfies δ ( G ) ≥ (1 − /r ) n contains a K r -factor. Moreover, there are n -vertex graphs G with δ ( G ) = (1 − /r ) n − that do not contain a K r -factor. More recently, K¨uhn and Osthus [9] determined, up to an additive constant, the minimum degreethreshold for forcing an H -factor, for any fixed graph H .The following result of Nenadov, Sudakov and Wagner [11] determines how many edges an n -vertex graph G needs to guarantee that G ∗ K t contains a K -factor (provided that t is not too bigcompared to n ). Theorem 1.3 (Nenadov, Sudakov and Wagner [11]) . There exists n ∈ N such that the followingholds. Let n, t ∈ N so that n ≥ n and | ( n + t ) , and let G be an n -vertex graph such that G ∗ K t does not contain a K -factor. If t ≤ n/ then e ( G ) ≤ (cid:18) n (cid:19) − (cid:18) k (cid:19) − ( k ( n − k ) if t is odd k ( n − k − if t is even,where k := ⌈ ( t + 1) / ⌉ . This bound on e ( G ) is sharp. A deficiency result for K r -factors. Nenadov, Sudakov and Wagner [11] state that ‘thestudy of deficiency concept by itself leads to intriguing open problems’ . In particular, the first openproblem [11, Section 7] they raise is to extend Theorem 1.3 to the full range of t , and moreover toresolve the analogous question for K r -factors in general. In this paper we fully resolve this problemvia the following theorem. Theorem 1.4.
Let n, t, r ∈ Z with n ≥ , t ≥ and r ≥ such that t < ( r − n and r | ( n + t ) .Further, let k := ⌈ t +1 r − ⌉ and q be the integer remainder when t is divided by r − . Let G be a graphon n vertices such that G ∗ K t does not contain a K r -factor. Then e ( G ) ≤ max (cid:26)(cid:18) n (cid:19) − (cid:18) n + tr + 12 (cid:19) , (cid:18) n (cid:19) − (cid:18) k (cid:19) − k ( n − k − ( r − − q )) (cid:27) . When ( r − | ( t + 1), the first term is at most the second term precisely when t ≤ ( r − n − r r − r +1 . Notethat Theorem 1.4 considers all interesting values of n and t . Indeed, if t ≥ ( r − n and r | ( n + t ), then G ∗ K t trivially contains a K r -factor, even if e ( G ) = 0. Further, in Section 3 we provide extremalexamples that demonstrate that the edge condition in Theorem 1.4 cannot be lowered. Perhapssurprisingly, the proof of Theorem 1.4 is short, making use of a couple of vertex-modification tricks(see the proofs of Lemmas 4.1 and 4.2) and Theorem 1.2.Note that the t = 0 case of Theorem 1.4 determines the edge density threshold for forcing a K r -factor in a graph. In fact, this is an old result due to Akiyama and Frankl [1], so our result canbe viewed as a deficiency generalisation of their theorem.1.2. A deficiency bandwidth theorem.
One of the central results in extremal graph theory isthe so-called
Bandwidth theorem due to B¨ottcher, Schacht and Taraz [2]. A graph H on n verticesis said to have bandwidth at most b , if there exists a labelling of the vertices of H with the numbers1 , . . . , n such that for every edge ij ∈ E ( H ) we have | i − j | ≤ b . Theorem 1.5 (The Bandwidth theorem, B¨ottcher, Schacht and Taraz [2]) . Given any r, ∆ ∈ N and any γ > , there exist constants β > and n ∈ N such that the following holds. Suppose that is an r -chromatic graph on n ≥ n vertices with ∆( H ) ≤ ∆ and bandwidth at most βn . If G isa graph on n vertices with δ ( G ) ≥ (cid:18) − r + γ (cid:19) n, then G contains a copy of H . Note that a K r -factor has bandwidth r −
1; thus, one can view the bandwidth theorem as a vastasymptotic generalisation of Theorem 1.2.Following the proof of Theorem 1.1 from [11], and applying a theorem of Knox and the thirdauthor [8], one can easily obtain a deficiency result for embedding bipartite graphs of boundeddegree and small bandwidth.
Theorem 1.6.
Given any ∆ ∈ N and ε > , there exist constants β > and n ∈ N such that thefollowing holds. Let t ∈ N and n ≥ n . Let H be a bipartite graph on n + t vertices with ∆( H ) ≤ ∆ and bandwidth at most β ( n + t ) . Suppose that G is a graph on n vertices such that G ∗ K t does notcontain a copy of H . Then we have the following bound on e ( G ) . e ( G ) ≤ (cid:18) n (cid:19) − (cid:0) t ( n − − (cid:0) t (cid:1) − εn (cid:1) if t ≤ n (cid:16)(cid:0) ⌈ n + t ⌉ +12 (cid:1) − εn (cid:17) if t > n . Observe that the bounds on e ( G ) in Theorem 1.6 are, up to error terms, exactly the sameas those in Theorem 1.1. Moreover, the extremal examples that show the condition on e ( G )in Theorem 1.1 is sharp also demonstrate that, for many graphs H , the condition on e ( G ) inTheorem 1.6 is asymptotically best possible (see Section 3). Notice the statement of Theorem 1.6is only interesting for t < n . Indeed, if t ≥ n then even if G has no edges, G ∗ K t contains all( n + t )-vertex bipartite graphs H (we just embed the smallest colour class into K t ).The paper is organised as follows. We introduce some graph theoretic notation in Section 2.In Section 3 we provide extremal examples for Theorems 1.4 and 1.6. In Section 4 we proveTheorem 1.4 and then in Section 5 we prove Theorem 1.6. Some concluding remarks are given inSection 6. 2. Notation
Let G be a graph. We define V ( G ) to be the vertex set of G and E ( G ) to be the edge set of G .Let X ⊆ V ( G ). Then G [ X ] is the graph induced by X on G and has vertex set X and edge set E ( G [ X ]) := { xy ∈ E ( G ) : x, y ∈ X } . For each x ∈ V ( G ), we define the neighbourhood of x in G tobe N G ( x ) := { y ∈ V ( G ) : xy ∈ E ( G ) } and define d G ( x ) := | N G ( x ) | .We write 0 < a ≪ b ≪ c < a, b, c from right to left.More precisely, there exist non-decreasing functions f : (0 , → (0 ,
1] and g : (0 , → (0 ,
1] suchthat for all a ≤ f ( b ) and b ≤ g ( c ) our calculations and arguments in our proofs are correct. Largerhierarchies are defined similarly.3. The extremal constructions for Theorem 1.4 and Theorem 1.6
In this section we will give the extremal constructions that match the upper bounds in Theo-rems 1.4 and 1.6. Firstly, let us consider those for Theorem 1.4.
Definition 3.1.
Let n, r, t ∈ Z with r ≥ , t ≥ and n ≥ such that t < ( r − n and r | ( n + t ) .Further, let k := ⌈ t +1 r − ⌉ and q be the integer remainder when t is divided by r − . We define graphs EX ( n, t, r ) and EX ( n, t, r ) as follows: Let K := K n and A ⊆ K such that A = K n + tr +1 . Define EX ( n, t, r ) to be the graphobtained by removing E ( A ) from K . • Consider a set of isolated vertices B where | B | = k and K n − k , and let C ⊆ V ( K n − k ) where | C | = r − − q . Define EX ( n, t, r ) to be the graph obtained by taking the disjoint union of B and K n − k and adding every edge incident to a vertex in C . Observe that e ( EX ( n, t, r )) = (cid:18) n (cid:19) − (cid:18) n + tr + 12 (cid:19) and e ( EX ( n, t, r )) = (cid:18) n (cid:19) − (cid:18) k (cid:19) − k ( n − k − ( r − − q )) . Hence max (cid:26)(cid:18) n (cid:19) − (cid:18) n + tr + 12 (cid:19) , (cid:18) n (cid:19) − (cid:18) k (cid:19) − k ( n − k − ( r − − q )) (cid:27) = max { e ( EX ( n, t, r )) , e ( EX ( n, t, r )) } . Next we show that EX ( n, t, r ) ∗ K t and EX ( n, t, r ) ∗ K t do not contain K r -factors, that is,they are extremal graphs for Theorem 1.4. Proposition 3.2. EX ( n, t, r ) ∗ K t and EX ( n, t, r ) ∗ K t do not contain K r -factors. Proof.
Firstly, let us consider EX ( n, t, r ) ∗ K t . For a contradiction, assume that EX ( n, t, r ) ∗ K t contains a K r -factor T . Then each vertex in V ( A ) belongs to a different copy of K r in T . Thisimplies n + tr = |T | ≥ | V ( A ) | = n + tr + 1, a contradiction. Hence EX ( n, t, r ) ∗ K t does not contain a K r -factor.Now let us consider EX ( n, t, r ) ∗ K t . For a contradiction, assume that EX ( n, t, r ) ∗ K t contains a K r -factor T . Then every vertex of B belongs to a different copy of K r in T . Thus, by construction,the copies of K r in T covering B must cover at least (cid:24) t + 1 r − (cid:25) · ( r −
1) = (cid:18) t − qr − (cid:19) · ( r −
1) = t − q + r − > t + | C | vertices in the copy of K t and C , a contradiction. Hence EX ( n, t, r ) ∗ K t does not contain a K r -factor. (cid:3) We now give the extremal constructions which, excluding error terms, match the upper boundsgiven in Theorem 1.6.
Definition 3.3.
Let n, t ∈ N such that ⌈ n + t ⌉ < n . We define graphs EX ( n, t ) and EX ( n, t ) asfollows: • Let K := K n and A ⊆ K such that A = K ⌈ n + t ⌉ +1 . Define EX ( n, t ) to be the graphobtained by removing E ( A ) from K . • Define EX ( n, t ) to be the disjoint union of a set of t isolated vertices and a clique of size n − t . One can see that the extremal examples in Definition 3.3 have the same construction to those inDefinition 3.1 for r = 2, except that in Definition 3.3 we omit the condition 2 | ( n + t ) and add thecondition that ⌈ n + t ⌉ < n (in order for EX ( n, t ) to be well-defined).Observe that e ( EX ( n, t )) and e ( EX ( n, t )) asymptotically match the upper bounds given inTheorem 1.6. Indeed, e ( EX ( n, t )) = (cid:18) n (cid:19) − (cid:18) ⌈ n + t ⌉ + 12 (cid:19) and e ( EX ( n, t )) = (cid:18) n (cid:19) − (cid:18) t ( n − − (cid:18) t (cid:19)(cid:19) . e conclude this section by showing that EX ( n, t ) ∗ K t and EX ( n, t ) ∗ K t do not containcertain ( n + t )-vertex bipartite graphs H . Definition 3.4.
Let H be the class of bipartite graphs H on n + t vertices with largest independentset of size ⌈ n + t ⌉ . Let H be the class of bipartite graphs H on n + t vertices which do not have atripartition ( A, B, C ) of V ( H ) such that | A | = n − t , | B | = | C | = t and every vertex in C is onlyadjacent to vertices in B . For example, the Hamilton cycle (when n + t is even) and K s,s -factors (for any fixed s ∈ N )belong to H ; the Hamilton cycle (when n + t is even) and K s,s -factors (for any fixed s ∈ N so that s does not divide t ) belong to H . Proposition 3.5. EX ( n, t ) ∗ K t does not contain any graph in H and EX ( n, t ) ∗ K t does notcontain any graph in H . Proof.
Firstly, let us consider EX ( n, t ) ∗ K t . Since EX ( n, t ) contains an independent set ofsize ⌈ n + t ⌉ + 1, any bipartite graph from H cannot be in EX ( n, t ).Now let us consider EX ( n, t ) ∗ K t . Since EX ( n, t ) has a set of t isolated vertices and | K t | = t ,we require that any bipartite graph H spanning EX ( n, t ) ∗ K t must have a tripartition ( A, B, C )of V ( H ) such that | A | = n − t , | B | = | C | = t and every vertex in C is only adjacent to verticesin B , where B = V ( K t ) and C is the set of t isolated vertices in EX ( n, t ). Hence EX ( n, t ) ∗ K t does not contain any graph in H . (cid:3) Proof of Theorem 1.4
The proof of Theorem 1.4 follows an inductive argument on the number of vertices n of G .Given a graph G such that G ∗ K t does not contain a K r -factor, we apply an appropriate vertex-modification procedure which, roughly speaking, allows us to assume G is locally isomorphic toone of the two extremal examples. This allows us to remove such local structure from G and applyinduction.The vertex-modification procedures are described by the following two structural lemmas re-garding graphs G with the property that G ∗ K t does not contain a K r -factor. Lemma 4.1 allowsus to assume that the degree of a vertex is either n − V ( EX ( n, t, r )) \ A and C ⊂ V ( EX ( n, t, r ))) or at most n − − ⌈ t +1 r − ⌉ (which is the degree of allvertices in V ( EX ( n, t, r )) \ ( B ∪ C )). Lemma 4.2 allows us to assume that either each edge of G belongs to some r -clique or there is a vertex with degree n − Lemma 4.1.
Let t ≥ , r ≥ and G be a graph on n vertices such that e ( G ) is maximal withrespect to the property that G ∗ K t does not contain a K r -factor. Then for every vertex v ∈ V ( G ) either d G ( v ) = n − or d G ( v ) ≤ n − − ⌈ t +1 r − ⌉ . Proof.
Suppose there exists a vertex v ∈ V ( G ) such that n − − (cid:24) t + 1 r − (cid:25) < d G ( v ) < n − . Let G ′ be the graph obtained from G by adding every possible edge incident to v , that is, d G ′ ( v ) = n −
1. Since e ( G ) is maximal with respect to G ∗ K t not containing a K r -factor, we must have that G ′ ∗ K t contains a K r -factor T ′ . Using T ′ , we will now construct a K r -factor T in G ∗ K t , givingus a contradiction.Let K v be the copy of K r in T ′ covering v . If K v ⊆ G ∗ K t then we can take T := T ′ . Henceassume K v * G ∗ K t . If there exists a copy K ′ of K r in T ′ that lies entirely in K t , then, for any u ∈ V ( K ′ ), we can take T := (cid:0) T ′ \ { K v , K ′ } (cid:1) ∪ { G [ { u } ∪ ( V ( K v ) \ { v } )] , G [ { v } ∪ (cid:0) V ( K ′ ) \ { u } (cid:1) ] } . ence assume no such copy of K r in T ′ exists. Since K v * G ∗ K t and no copy of K r in T ′ liesentirely in K t , the number of copies of K r in T ′ which cover some vertex of { v } ∪ V ( K t ) in G ′ ∗ K t isat least ⌈ t +1 r − ⌉ . But d G ( v ) > n − − ⌈ t +1 r − ⌉ , hence there exists a copy ˆ K of K r in T ′ which intersects { v } ∪ V ( K t ) and whose vertices are all neighbours of v in G ∗ K t or v itself. Now, if v ∈ V ( ˆ K ), then K v = ˆ K ⊆ G ∗ K t , a contradiction to our previous assumption. Hence V ( ˆ K ) ∩ V ( K t ) = ∅ and, forany u ∈ V ( ˆ K ) ∩ V ( K t ), we can take T := ( T ′ \ { K v , ˆ K } ) ∪ { G [ { u } ∪ ( V ( K v ) \ { v } )] , G [ { v } ∪ ( V ( ˆ K ) \ { u } )] } . (cid:3) Lemma 4.2.
Let t ≥ , r ≥ . Let G be a graph on n vertices such that G ∗ K t does not containa K r -factor and suppose G contains an edge which is not contained in any copy of K r in G . Thenthere exists a graph G ′ on n vertices such that G ′ ∗ K t does not contain a K r -factor, e ( G ) ≤ e ( G ′ ) and G ′ has a vertex of degree n − . Proof.
Let xy be an edge in G which is not contained in any copy of K r . Let Q be a clique ofmaximal size containing xy and set ℓ := | V ( Q ) | . Observe that every vertex in G has at most ℓ − Q as otherwise xy would lie in an ( ℓ + 1)-clique. Thus X v ∈ V ( Q ) d G ( v ) ≤ n ( ℓ − . Let G ′ be the graph obtained by deleting all edges between x and vertices in V ( G ) \ V ( Q ) and,subsequently, adding any missing edge incident to any vertex in V ( Q ) \ { x } . Note that Q is stillan ℓ -clique in G ′ and X v ∈ V ( Q ) d G ′ ( v ) = n ( ℓ − . Hence e ( G ) ≤ e ( G ′ ). Also, G ′ is a graph on n vertices and d G ′ ( y ) = n −
1. It remains to show that G ′ ∗ K t does not contain a K r -factor. Suppose, for a contradiction, that G ′ ∗ K t does contain a K r -factor T ′ . We will now use T ′ to construct a new K r -factor T in G ′ ∗ K t which does not containany edge vw where v ∈ V ( Q ) and w ∈ V ( G ′ ) \ V ( Q ). Such a K r -factor T is also a K r -factor in G ∗ K t , giving us a contradiction.Suppose that the copy K x of K r in T ′ covering x has exactly s vertices in K t . Then K x hasexactly r − s vertices in Q . Thus there are ℓ − r + s remaining vertices in V ( Q ) \ V ( K x ). Notethat ℓ − r + s ≤ s , hence there is an injection f : V ( Q ) \ V ( K x ) → V ( K x ) ∩ V ( K t ). We constructour K r -factor T as follows: for every copy of K r in T ′ intersecting Q other than K x , we substituteall vertices lying in its intersection with Q with their images under f . Finally, we take the copyof K r formed by the ℓ -clique Q and the s − ( ℓ − r + s ) = r − ℓ vertices in V ( K x ) ∩ V ( K t ) whichdo not appear in the image of f . Observe that T does not use any edge vw where v ∈ V ( Q ) and w ∈ V ( G ) \ V ( Q ), and we are done. (cid:3) Before proceeding to the proof of Theorem 1.4, we prove the following technical lemma.
Lemma 4.3.
Let n, t, r ∈ N such that r ≥ ; r − divides t + 1 ; r divides n + t ; (1) e ( EX ( n − , t + 1 , r )) < e ( EX ( n − , t + 1 , r )) . Then (2) e ( EX ( n − , t + 1 , r )) + ( n − ≤ e ( EX ( n, t, r )) . So here we are implicitly assuming that the choice of n, t, r are such that EX ( n − , t +1 , r ) and EX ( n − , t +1 , r )are well-defined. roof. By considering how to ‘move’ from EX ( n − , t + 1 , r ) to EX ( n, t, r ) one obtains that e ( EX ( n, t, r )) − e ( EX ( n − , t + 1 , r )) = − ( r − (cid:18) t + 1 r − (cid:19) + (cid:18) n − t + 1 r − − r (cid:19) + (cid:18) n − t + 1 r − − (cid:19) . Rearranging this, one obtains that (2) holds precisely when t ≤ n − r − nr =: g ( n, r ) . Further, one can calculate that e ( EX ( n − , t + 1 , r )) ≤ e ( EX ( n − , t + 1 , r )) precisely when(3) f ( n, r ) := n ( r − r − r + 1) − r ≤ t ≤ n ( r − − r =: f ( n, r ) . Since (1) holds, we have that (3) implies that t < f ( n, r ) or t > f ( n, r ). Observe that f ( n, r ) ≤ g ( n, r ). Thus, if t < f ( n, r ) then t < g ( n, r ) and the claim holds.Suppose t > f ( n, r ). We will show that in this case the hypothesis of the lemma cannot actuallyhold. In particular, under the assumptions that r − t + 1, r divides n + t and t > f ( n, r ), EX ( n − , t + 1 , r ) is undefined. Indeed, for a contradiction let us assume that EX ( n − , t + 1 , r )is well-defined in this case, thus the inequality t + 1 < ( r − n −
1) must hold. By assumption, t satisfies the following modular equations:(4) t ≡ − r −
1) and t ≡ − n (mod r ) . For n and r fixed, the solution of (4) is unique modulo r ( r −
1) by the Chinese Remainder Theorem,since r and r − t ′ = ( r − n − − t = ( r − n − − − kr ( r −
1) for some integer k . The constraint t + 1 < ( r − n −
1) forces k ≥
1, hence t ≤ ( r − n − − − r ( r −
1) = n ( r − − r = f ( n, r ) . This contradicts the assumption that t > f ( n, r ). (cid:3) Proof of Theorem 1.4.
We prove Theorem 1.4 by induction on n . If n = 2 then e ( G ) ∈ { , } .Since t < r −
1) and r | ( n + t ), we must have t = r −
2. If e ( G ) = 1 then G ∗ K t is a copy of K r ,contradicting our choice of G . Thus e ( G ) = 0. Observe that k = ⌈ t +1 r − ⌉ = 1 and q = r −
2. Thusmax (cid:26)(cid:18) n (cid:19) − (cid:18) n + tr + 12 (cid:19) , (cid:18) n (cid:19) − (cid:18) k (cid:19) − k ( n − k − ( r − − q )) (cid:27) = max { , } = 0 . Thus Theorem 1.4 holds for n = 2.For the inductive step, we may assume without loss of generality that G is a graph on n verticessuch that e ( G ) is maximal with respect to the property that G ∗ K t does not contain a K r -factor. Case (i): G contains an isolated vertex v . If t < r −
2, then one could add a single edge to v and G would still not contain a K r -factor, contradicting our choice of G . Hence t ≥ r −
2. If t = r − k = ⌈ t +1 r − ⌉ = 1 and q = r −
2. Hencemax (cid:26)(cid:18) n (cid:19) − (cid:18) n + tr + 12 (cid:19) , (cid:18) n (cid:19) − (cid:18) k (cid:19) − k ( n − k − ( r − − q )) (cid:27) = max (cid:26)(cid:18) n (cid:19) − (cid:18) n − r + 22 (cid:19) , (cid:18) n − (cid:19)(cid:27) . Since G contains at least one isolated vertex, e ( G ) ≤ (cid:18) n − (cid:19) ≤ max (cid:26)(cid:18) n (cid:19) − (cid:18) n − r + 22 (cid:19) , (cid:18) n − (cid:19)(cid:27) , nd we are done. If t ≥ r −
1, consider the graph G ′ obtained by deleting v from G . Since G ∗ K t does not contain a K r -factor, G ′ ∗ K t − r +1 does not contain a K r -factor. Thus, by our inductivehypothesis e ( G ′ ) ≤ max { e ( EX ( n − , t − r + 1 , r )) , e ( EX ( n − , t − r + 1 , r )) } . It follows from e ( G ) = e ( G ′ ) that e ( G ) ≤ max { e ( EX ( n, t, r )) , e ( EX ( n, t, r )) } . Case (ii): G contains a vertex of degree n − . Consider the graph G ′ obtained by deleting sucha vertex from G . Note that G ∗ K t = G ′ ∗ K t +1 . If t + 1 ≥ ( r − n − G ′ ∗ K t +1 contains a K r -factor, a contradiction. So we must have that t + 1 < ( r − n −
1) and hence byinduction e ( G ′ ) ≤ max { e ( EX ( n − , t + 1 , r )) , e ( EX ( n − , t + 1 , r )) } . Observe that e ( G ) = e ( G ′ ) + n −
1. We aim to show that(5) e ( G ) ≤ max { e ( EX ( n, t, r )) , e ( EX ( n, t, r )) } . If e ( G ′ ) ≤ e ( EX ( n − , t + 1 , r ) then e ( G ∗ K t ) = e ( G ′ ∗ K t +1 ) ≤ e ( EX ( n − , t + 1 , r ) ∗ K t +1 ) = e ( EX ( n, t, r ) ∗ K t )and thus e ( G ) ≤ e ( EX ( n, t, r )) . Similarly, if e ( G ′ ) ≤ e ( EX ( n − , t + 1 , r ) and r − t + 1 then e ( G ∗ K t ) = e ( G ′ ∗ K t +1 ) ≤ e ( EX ( n − , t + 1 , r ) ∗ K t +1 ) = e ( EX ( n, t, r ) ∗ K t )and thus e ( G ) ≤ e ( EX ( n, t, r )) . It remains to check that (5) holds in the case when ( r − | ( t + 1) and e ( EX ( n − , t + 1 , r )) < e ( EX ( n − , t + 1 , r )) . In this case Lemma 4.3 implies that e ( G ) = e ( G ′ ) + n − ≤ e ( EX ( n − , t + 1 , r )) + ( n − ≤ e ( EX ( n, t, r )) , as desired. Case (iii): G contains no vertex of degree or n − . If G contains an edge which is not containedin any copy of K r , then by Lemma 4.2 there exists a graph G ′ on n vertices such that G ′ ∗ K t doesnot contain a K r -factor, e ( G ) ≤ e ( G ′ ) and G ′ has a vertex of degree n −
1. The argument fromCase (ii) then implies that e ( G ) ≤ e ( G ′ ) ≤ max { e ( EX ( n, t, r )) , e ( EX ( n, t, r )) } .We may therefore assume every edge of G is contained in some copy of K r . Moreover, as novertex in G has degree 0, every vertex in G is contained in a copy of K r . Let w be a vertex ofsmallest degree in G . If d G ( w ) ≥ ( r − n − tr then δ ( G ∗ K t ) ≥ ( r − n − tr + t = r − r ( n + t ) . Hence by Theorem 1.2, we have that G ∗ K t contains a K r -factor, a contradiction.Thus d G ( w ) < ( r − n − tr . Let K be a copy of K r in G containing w . Consider the graph G ′ obtained by removing K and all its vertices from G . Then G ′ ∗ K t does not contain a K r -factor;this implies that t < ( r − n − r ). Hence, by our inductive hypothesis, we have e ( G ′ ) ≤ max { e ( EX ( n − r, t, r )) , e ( EX ( n − r, t, r )) } . If e ( G ′ ) ≤ e ( EX ( n − r, t, r )) then d G ( w ) < ( r − n − tr implies e ( G ) ≤ e ( EX ( n, t, r )), as desired.By Lemma 4.1, we have that ∆( G ) ≤ n − − ⌈ t +1 r − ⌉ since we assumed G contains no vertices ofdegree n − e ( G ) is maximal with respect to the property that G ∗ K t does not contain a r -factor. Thus, if e ( G ′ ) ≤ e ( EX ( n − r, t, r )) then applying this observation yields that e ( G ) ≤ e ( EX ( n, t, r )), as desired. (cid:3) Proof of Theorem 1.6
In the proof of Theorem 1.6 we will make use of the following theorem of Knox and Treglown [8].
Theorem 5.1 (Knox and Treglown [8]) . Given any ∆ ∈ N and any γ > , there exists constants β > and n ∈ N such that the following holds. Suppose that H is a bipartite graph on n ≥ n vertices with ∆( H ) ≤ ∆ and bandwidth at most βn . Let G be a graph on n vertices with degreesequence d ≤ · · · ≤ d n . If d i ≥ i + γn for all i < n/ then G contains a copy of H . In fact, Knox and Treglown proved a more general result for robust expanders (see [8, Theo-rem 1.8]). We use Theorem 5.1 in a similar way to how Chv´atal’s theorem [4] is used in the proofof Theorem 1.1 in [11].
Proof.
Let ∆ ∈ N and ε >
0. Define γ > γ ≪ ε, / ∆. Apply Theorem 5.1 with ∆and γ to produce constants β > n ∈ N such that0 < n ≪ β ≪ γ ≪ ε, . Let t ∈ N , n ≥ n and H be an ( n + t )-vertex graph as in the statement of the theorem. Supposethat G is a graph on n vertices such that G ∗ K t does not contain H . Let m ( G ) denote the numberof missing edges in G , that is, m ( G ) := (cid:0) n (cid:1) − e ( G ). Then proving Theorem 1.6 is equivalent toproving the following bound on m ( G ): m ( G ) ≥ t ( n − − (cid:0) t (cid:1) − εn if t ≤ n (cid:0) ⌈ n + t ⌉ +12 (cid:1) − εn if t > n . (6)Let G ′ := G ∗ K t and label the vertices of G ′ as v , . . . , v n + t such that d G ′ ( v i ) := d i is the degreeof vertex v i and d ≤ d ≤ . . . ≤ d n + t . Since G ′ does not contain H as a subgraph, it does notsatisfy the degree sequence condition of Theorem 5.1. Moreover, δ ( G ′ ) ≥ t , hence there must exist t − γ ( n + t ) < i ≤ ⌈ ( n + t ) / ⌉ − d i < i + γ ( n + t ). From d ≤ . . . ≤ d i < i + γ ( n + t ) wededuce that the number of edges missing from G ′ is at least m ( G ′ ) ≥ i X j =1 ( n + t − − d j ) − (cid:18) i (cid:19) > i ( n + t − − i − γ ( n + t )) − (cid:18) i (cid:19) ≥ i ((1 − γ )( n + t ) − i ) − (cid:18) i (cid:19) =: f ( i ) . (7)Set u := ⌈ ( n + t ) / ⌉ −
1. Now f ( i ) is a quadractic in i and d ( f ( i )) di <
0. Also, note that m ( G ) = m ( G ′ ). Hence, as t − γ ( n + t ) < i ≤ u , we have from (7) that(8) m ( G ) ≥ min { f ( t − γ ( n + t )) , f ( u ) } . One can calculate that(9) f ( t − γ ( n + t )) ≤ f ( u ) if and only if t ≤ n − γn +85+2 γ if n + t is even n − γn +55+2 γ if n + t is odd . s n ≪ γ ≪ ε we have(10) f ( t − γ ( n + t )) ≥ t ( n − − (cid:18) t (cid:19) − εn and(11) f ( u ) ≥ (cid:18) ⌈ n + t ⌉ + 12 (cid:19) − εn . Moreover, for n − γn +55+2 γ ≤ t ≤ n we have(12) f ( u ) ≥ (cid:18) ⌈ n + t ⌉ + 12 (cid:19) − εn ≥ t ( n − − (cid:18) t (cid:19) − εn . Regardless of the parity of n + t , using (8)–(12) we conclude that (6) holds. (cid:3) Concluding remarks
In this paper we resolved the deficiency problem for K r -factors. For a general fixed graph H , itwould be interesting to prove deficiency results regarding H -factors. As a starting point for thisproblem we pose the following question. Let α ( H ) denote the size of the largest independent setin H . Question 6.1.
Let K := K n and A ⊆ K such that A = K α ( H )( n + t ) | H | +1 . Define EX H ( n, t ) to bethe graph obtained by removing E ( A ) from K . Does there exist a constant c := c ( H ) > suchthat if t ≥ cn and G is an n -vertex graph so that G ∗ K t does not contain an H -factor then e ( G ) ≤ e ( EX H ( n, t )) + o ( n ) ? Note that Theorem 1.6 answers this question in the affirmative e.g. for H = K s,s (for fixed s ∈ N ). On the other hand, at least for some H one cannot remove the o ( n ) term in Question 6.1completely. Indeed, let H = K ,s where s ≥ n -vertex graph EX ′ H ( n, t ) obtainedfrom EX H ( n, t ) by adding a maximal matching in V ( A ). It is easy to see that EX ′ H ( n, t ) ∗ K t does not contain a K ,s -factor. This example suggests it might be rather challenging to resolve the H -factor deficiency problem completely for all graphs H .It would also be interesting to prove bandwidth deficiency results in the vein of Theorem 1.6 fornon-bipartite graphs H . References [1] A. Akiyama and P. Frankl, On the Size of Graphs with Complete-Factors,
J. Graph Theory (1985), 197–201.[2] J. B¨ottcher, M. Schacht and A. Taraz, Proof of the bandwidth conjecture of Bollob´as and Koml´os, Math.Ann. (2009), 175–205.[3] D. Bryant and D. Horsley, A proof of Lindner’s conjecture on embeddings of partial Steiner triple systems,
J.Combin. Des. (2009), 63–89.[4] V. Chv´atal, On Hamilton’s ideals, J. Combin. Theory Ser. B (1972), 163–168.[5] D. Daykin and R. H¨aggkvist, Completion of sparse partial Latin square, in: Proc. Cambridge Conference inHonour of Paul Erd˝os, 1983.[6] T. Evans, Embedding in complete Latin squares, Am. Math. Mon. (1960), 958–961.[7] A. Hajnal and E. Szemer´edi, Proof of a conjecture of Erd˝os, Combinatorial Theory and its Applications vol. II (1970), 601–623.[8] F. Knox and A. Treglown, Embedding spanning bipartite graphs of small bandwidth, Combin. Probab. Comput. (2013), 71–96.[9] D. K¨uhn and D. Osthus, The minimum degree threshold for perfect graph packings, Combinatorica (2009),65–107.
10] C.C. Lindner, A partial Steiner triple system of order n can be embedded in a Steiner triple system of order6 n + 3, J. Combin. Theory Ser. A (1975), 349–351.[11] R. Nenadov, B. Sudakov and A.Z. Wagner, Completion and deficiency problems, J. Combin. Theory Ser. B (2020), 214–240.Andrea Freschi, Joseph Hyde & Andrew TreglownSchool of MathematicsUniversity of BirminghamBirminghamB15 2TTUK
E-mail addresses: { axf079, jfh337, a.c.treglown } @[email protected]