Regarding two conjectures on clique and biclique partitions

For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures -- one regarding the maximum possible value of $cp(G) + cp(\overline{G})$ (due to de Caen, Erd\H{o}s, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $\max_G cp(G) + cp(\overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.


Introduction
For a fixed family of graphs F , an F -partition of a graph G is a collection C = {H 1 , . . . , H ℓ } of subgraphs H i ⊂ G such that each edge of G belongs to exactly one H i ∈ C, and each H i is isomorphic to some graph in F . When F = {K r } r≥2 , we refer to F -partitions as clique partitions, and when F = {K s,t } s,t≥1 , the corresponding partitions are called biclique partitions. The size |C| of the smallest clique partition of G is called the clique partition number of G, denoted cp(G). The biclique partition number bp(G) is defined analogously. Both cp(G) and bp(G) (and their many variants) are NP-hard to compute in general graphs, but have been studied extensively from a combinatorial perspective, in part because of their connections to various areas of computer science (see, e.g. [13]). In this paper, we consider two longstanding combinatorial questions related to these quantities.

Biclique partitions of K n
In 1971, Graham and Pollak [9] showed that, for every n ≥ 2, bp(K n ) = n − 1. (1.1) In particular, the edges of K n can be partitioned into n − 1 stars . As a matter of notation, such a collection is called a k-biclique cover of G. More generally, a {k 1 , . . . , k t }-biclique cover of G is a collection {H 1 , . . . , H ℓ } of bicliques H i ⊂ G such that for each edge of G there is some k ∈ {k 1 , . . . , k t } such that the edge belongs to exactly k of the bicliques. In 1993, de Caen, Gregory and Pritikin conjectured that (1.2) is tight for sufficiently large n: Conjecture 1.1 (de Caen et al. [3]). For every positive integer k, bp k (K n ) = n − 1 for all sufficiently large n.
The same authors prove their conjecture for each k ≤ 18, using special constructions from design theory [3]. However, the best-known upper bound for general k is bp k (K n ) = O(kn), obtained by simply compounding a small-k construction.
In Section 2, we show that, to leading order, Conjecture 1.1 is true. More precisely, we construct a family of designs (inspired by classical ideas of Nisan and Wigderson [15]), that yields a k-covering of K n by at most n + 2kn 3/4 + k √ n complete bipartite subgraphs.

Clique partitions of G and G
In 1986, de Caen, Erdős, Pullman and Wormald [4] investigated the maximum value of cp(G) + cp(G) over the set G n of all graphs G on n vertices, and proved that In Section 3.1, we show that the family of graphs constructed in [4] can actually be modified to improve the lower bound in (1.3), thereby disproving Conjecture 1.3. Theorem 1.4. For infinitely many n, there exists a self-complementary graph G ∈ G n with cp(G) ≥ 23 164 n 2 + o(n 2 ). The upper bound in (1.3) essentially comes from greedily selecting edgedisjoint triangles from G and G, forming clique partitions into K 3 's and K 2 's. Subsequent work on complementary triangle packings, first by Erdős et al. [7] and later by Keevash and Sudakov [14], improved significantly upon the greedy packing, with the latter authors showing the existence of a packing with n 2 12.89 edge-disjoint triangles. The resulting clique partitions (as observed by Bujtas et al. [2]) contain a total of 0.34481n 2 + o(n 2 ) cliques, improving the 0.43n 2 upper bound in (1.3). However, partitions into triangles and edges can never push this bound below 0.33n 2 , as illustrated by G = K n/2,n/2 . In Section 3.2, we extend the ideas of Keevash and Sudakov to the complementary clique partition problem, improving (1.3) beyond the limits of triangle packings: Theorem 1.5. For all G ∈ G n , cp(G) + cp(G) ≤ 0.3186n 2 + o(n 2 ). 1 In the same paper [4], the authors solve the corresponding problem for cc(G) + cc(G), where cc(G) is the minimal number of cliques in G needed to cover every edge at least once, showing that max G∈Gn cc(G) + cc(G) = n 2 4 (1 + o (1)). This is tight up to the o(1) error by K n/2,n/2 , and the error term was later removed by Pyber [19] for n > 2 1500 .

A k-biclique covering of K n
Our goal in this section is to construct a collection of (1 + o(1))n bicliques on a set of n vertices such that all n 2 edges belong to exactly k bicliques in the collection. We recall the definition of a combinatorial design in the sense of Nisan and Wigderson [15] from their classical paper on pseudorandom generators.
We construct our designs in a way that differs from [15] and better suits our particular choice of parameters: For i ∈ [n], let S i consist of m elements from U, one from each group. Specifically, for k ∈ [m], pick element i (mod p k ) from group Z/p k Z.
It is clear that |S i | = m for all i, and that d := |U| = m k=1 p k ≤ 2mn 1/(t+1) . We claim that |S i ∩ S j | ≤ t for all distinct i, j ∈ [n]. Indeed, suppose to the contrary that |S i ∩ S j | > t for some distinct i, j ∈ [n]. Then among the chosen primes, there are t + 1 primes p l 1 , . . . , p l t+1 with i ≡ j (mod p l k ) for each k ∈ [t + 1]. But then t+1 k=1 p l k (i − j).
Remark 2.3. The above design is in fact optimal up to constant factors. Consider any (n, d, t, m)-design, where the sets are contained in a universe U of size d. For every (t + 1)-element subset of U, there is at most one set among S 1 , . . . , S n that contains the subset. Since each S i contains m t+1 subsets of size t + 1, we must have d t+1 ≥ n m t+1 , so We will only use the special case (n, k √ n, 1, ⌊k/2⌋) of Lemma 2.2, which we state explicitly below as a corollary.
We also require a result of Alon [1] on {1, 2}-biclique coverings of K n , which are collections of bicliques such that every edge of K n belongs to either 1 or 2 of the bicliques in the collection. The size of the smallest such collection is denoted bp {1,2} (K n ). Finally, we construct a k-biclique covering of K n .
Then any edge {i, j} is covered exactly |S i | + |S j | − 2|S i ∩ S j | times, and this number is equal to either 2⌊k/2⌋ or 2⌊k/2⌋ − 2 (depending on whether |S i ∩ S j | = 0 or 1). If k is odd, every edge still needs to be covered either 1 or 3 more times. Let us define a triple-edge to be an edge {i, j} with |S i ∩ S j | = 1. An edge {i, j} is a triple-edge if and only if there exists some index l and remainder r such that i ≡ j ≡ r (mod p l ). We can define a clique C l,r consisting of all vertices i with i ≡ r (mod p l ). Observe that every triple-edge is contained in exactly one such clique, and every such clique contains only triple-edges. To make progress, we will construct a {1, 2}-biclique covering of each clique C l,r . The number of cliques C l,r is at most k √ n, and each has size at most √ n, so by Fact 2.5, at most k √ n · 2n 1/4 = 2kn 3/4 bicliques are needed to {1, 2}-cover every clique C l,r . Now every edge needs to be covered 1 or 2 more times.
If k is even, every edge needs to be covered only 0 or 2 more times, so we skip the above step. Finally, in either case, we'll "pad" the covering so that every edge is covered exactly k times. To do this, define bicliques D 1 , . . . , D n where D i is the star centered at vertex i and containing edges to all vertices j < i such that {i, j} needs to be covered 1 or 2 more times, and to all vertices j > i such that {i, j} needs to be covered 2 more times.
This completes the construction. The total number of bicliques used is at most n + 2kn 3/4 + k √ n (from the padding step, the {1, 2}-covering step, and the initial design, respectively).
Remark 2.7. A key ingredient in the proof above is the {2k − 2, 2k}-biclique covering of K n using 2k √ n bicliques. It is shown in [5] that n/2 bicliques are necessary for this list covering, so the asymptotic dependence on n cannot be decreased.

Improving the lower bound
The construction in our proof of Theorem 1.4 is based on the original construction in [4], and the calculation of its clique partition number makes use of certain facts shown in [4] and [17]. Here we include the entire argument for the reader's convenience. Before proceeding with the construction, we need the following lemma, which has appeared in many places but perhaps first in Pullman and Donald [17]. Recall that the edge chromatic number χ ′ (G) of a graph G is the minimum number of colors needed to color the edges of G so that no two edges of the same color are incident to the same vertex. We use the notation G ≡ H to denote the graph on vertices V (G) ⊔ V (H) formed by adding all edges between V (G) and V (H).
Proof. Let H = K ℓ and let E G−H be the set of all nℓ edges between V (G) and When χ ′ (G) ≤ ℓ, we can assign each of the ℓ nodes in H to one of the ℓ color classes of a valid edge coloring in G, and obtain a collection of triangles of the form {v, x, y}, for v ∈ H and (x, y) ∈ E(G) that has been given color v in the edge coloring. No edge in E G−H will be used twice precisely because no vertex in G is incident to two edges of the same color. This gives a collection of e edge-disjoint triangles that cover all the edges in G, and leaves at most nℓ − 2e edges left to cover. Adding in those remaining edges yields a clique partition of size at most nℓ − e.
The construction: Let ℓ and m be any positive integers, and let G be any graph on m vertices. We define H ℓ = H ℓ (G) to be the graph in Figure Figure 1: The graph H ℓ (G) and its complement.
1, where the double lines are to be interpreted in the same way as the ≡ symbol, i.e. including all possible edges between the vertices on either end.
Observe that H ℓ (G) ∼ = H ℓ (G), and that the edges of H ℓ (G) can be split into , which is at most m, since we can assign the numbers 0, 1, . . . , m − 1 to each vertex and color the edge for any graph G on m ≤ 2ℓ vertices. (In fact, this still gives a lower bound on cp(H ℓ (G)) + cp(H ℓ (G)) for any G and any m.) The term cp(Y ℓ ) was computed in [4], and we include this calculation in the Appendix: So for any G on m vertices, we have Note that H ℓ (G) has n := m + 4ℓ vertices, so when we maximize (3.2) in m while keeping n fixed, we find that the optimum occurs at m = 9 8 ℓ. At this value of m, the lower bound is (8 − 81 128 )ℓ 2 + O(ℓ) for a graph on 41 8 ℓ vertices, implying that, for infinitely many n, Note that if G is a self-complementary graph (i.e. G ∼ = G), then H ℓ (G) is also self-complementary.

Improving the upper bound
The problem of partitioning a graph G into as few cliques as possible is equivalent to the problem of packing disjoint copies of K 3 , K 4 , . . . , K n inside of G in such a way as to maximize a certain linear objective function. Indeed, given a clique partition C of G, let C i denote the number of cliques of size i in C, for i = 2, . . . , n. Then |C| = n i=2 C i and n i=2 i We will also consider r-restricted clique packings/partitions, in which the largest clique can have size at most r. We define cp(G, r) to be the minimum number of cliques of size at most r needed to partition the edges of G.
Clearly cp(G, r) ≥ cp(G), and one would expect the numbers cp(G, r) and cp(G) to be relatively close for large r. This is indeed the case, as we show in the following lemma.

Lemma 3.3.
For any ǫ > 0, there exists an integer r 0 = r 0 (ǫ) such that for any r ≥ r 0 and any graph G on n vertices, Proof. We make use of the following fact, which is a straightforward consequence of Wilson's theorem [21]: for any fixed t ≥ 2 and ǫ > 0, there is an integer m 0 = m 0 (t, ǫ) such that for all m ≥ m 0 , there is a partition of K m into edge-disjoint copies of K t and at most ǫm 2 leftover edges. Set t = 1 2ǫ and r 0 = m 0 (t, ǫ/5).
Let C be a clique partition with |C| = cp(G). For any r ≥ r 0 , we can obtain an r-restricted clique partition C from C as follows: keep each clique of size at most r, and, for each clique K m with m > r, decompose it into at most m 2 / t 2 copies of K t and cover the remaining edges (of which there are at most ǫ 5 · m 2 ) with K 2 's. This gives a clique partition C of size from which the lemma follows.

Fractional clique packings
For a fixed family F of graphs and any graph G, let G F denote the set of (unlabeled, non-induced) subgraphs of G which are isomorphic to some F ∈ F . Following Keevash and Sudakov [14] and Yuster [22], we say a function ψ : G We denote by G F the polyhedron of all fractional F -packings of G. As we are interested in the fractional analogue of clique packings, we will only be concerned with families of the form When the objective function is simply H∈( G F ) ψ(H), and the family F = {F } is just a single graph, a theorem of Haxell and Rödl [11] implies that relaxing the domain of maximization from (integer) packings to fractional packings can only change the value of the optimum by o(n 2 ). Subsequently, Yuster [22] extended this result to arbitrary families of graphs. For finite families (such as F r ), Yuster's proof easily extends to arbitrary linear objective functions [23]. Therefore: The advantages of studying fractional clique packings rather than clique partitions are twofold. First, solving the linear program (3.5) is computationally feasible, unlike the corresponding integer program. Second, they can be averaged, which not only enables one to turn finite computations into asymptotic bounds, but also allows one to leverage the results of a search on n vertices to reduce the search space when looking for a minimizer on n + 1 vertices. This is the approach used by Keevash and Sudakov in [14], and the following averaging lemma (for a different LP) appears as their Lemma 2.1, with the same proof.

Lemma 3.5. For any r ≥ 3, the sequence fr(n) n(n−1) is increasing in n.
Proof. Let G ∈ G n+1 , and let G 1 , . . . , G n+1 be the induced subgraphs on the vertex subsets of size n. Let ψ i , ψ i be optimal fractional packings on G i and G i . Since each edge of G (and G) occurs in n − 1 of the G i , we have that are fractional packings on G and G with combined objective value of at least n+1 n−1 f r (n), and hence fr(n+1) (n+1)n ≥ (n+1)fr (n) n(n−1)(n+1) = fr(n) n(n−1) , as claimed.
Since the sequence fr(n) n(n−1) is obviously bounded above by 1/2, it follows that it converges to a limit c r ∈ (0, 1/2). Since c r is increasing in r, the sequence {c r } also converges to a limit that we will call c ∞ .
Proof. This essentially follows from Lemma 3.3 and Theorem 3.4. More explicitly, for any ǫ > 0, we can pick r large enough so that |cp(G)−cp(G, r)| < ǫn 2 for any G ∈ G n , and |c r − c ∞ | < ǫ. Now pick n large enough so that |v r (G) − ν r (G)| < ǫn 2 for any G ∈ G n and |f r (n) − c r n 2 | < ǫn 2 . It follows that max for n sufficiently large.
The same argument shows that max G∈Gn cp(G, r)+cp(G, r) ∼ 1 2 − c r n 2 . Let us define α r := 1 2 − c r , and α ∞ = 1 2 − c ∞ . We seek an upper bound on α ∞ , and since α ∞ ≤ α r = 1 2 − c r ≤ 1 2 − fr(n) n(n−1) for any n, it suffices for our purposes to compute a lower bound on the value of fr(n) n(n−1) for any particular pair of positive integers (r, n). For example, a modern computer can compute f 4 (8) = 6 numerically by solving the LP (3.5) on every non-isomorphic graph on 8 vertices. This shows that α ∞ ≤ α 4 ≤ 1 2 − 6 8·7 = 11 28 ≈ 0.3928. This already beats the best bound one can get from purely Ramsey-based arguments 2 , although it does not beat the Keevash-Sudakov triangle packing bound. In the remainder of this section, we improve this bound in two ways: first, we show in Section 3.2.2, we can combine Ramsey-type arguments with estimates on f r (n 0 ) to yield better estimates on f r (n) for n much larger than n 0 ; second, in Section 3.2.3 we compute the exact value of f 4 (n) up to n = 19, 2 As was remarked in [4], one can begin with a maximal collection of edge disjoint K r 's (instead of triangles) in G and G, and bound the number of remaining edges (using Turan's theorem) by ξ r n 2 , where ξ r := 1 2 − 1 2R(r,r)−2 , and the iterate on the remaining edges with cliques of size K r−1 , etc. It is not hard to see that the bound one obtains is Even using the most optimistic (i.e. smallest) of the possible values for R(k, k) for k ≥ 5, this approach will not yield an upper bound better than 0.41n 2 .
using an algorithm of Keevash and Sudakov that is significantly more efficient than brute force search.

Ramsey-type improvements
In [14], it was observed that the averaging argument in Lemma 3.5 can be improved, in a sense, by using a different decomposition of G into smaller subgraphs based on a greedy packing as described in the introduction. In particular, given any bicoloring of K 3n , greedily select vertex-disjoint monochromatic triangles T 1 , . . . , T i . The fact that R(3, 3) = 6 guarantees that we can do this until 3 vertices remain, giving us n − 1 triangles T 1 , . . . , T n−1 , and one set of 3 vertices denoted T n . Consider the 3 n colorings c of K n obtained by picking one vertex in each T i and the edges between them. Each coloring has some fractional packing ψ c of weight at least f 3 (n), and since each edge between T i and T j for i = j occurs in exactly 3 n−2 of these, the average 3 −(n−2) c ψ c is a valid fractional packing in K 3n of weight at least 9f 3 (n). Since each of the monochromatic triangles T 1 , . . . , T n−1 are edge disjoint from this packing, they can be included as well, yielding a lower bound Since R(4, 4) = 18, we can greedily find vertex disjoint monochromatic copies of K 4 , H 1 , . . . , H n−4 , with 16 vertices remaining. From the remaining vertices, we can find edge disjoint monochromatic triangles T n−3 , T n−2 , T n−1 , T n , which we join with the remaining four vertices to form H n−3 , . . . , H n , each of size four. Repeating the same process as above, we see that For r = 5, we can use the bound R(5, 5) ≤ 48 to find n − 9 vertex-disjoint copies of K 5 , with 45 vertices left over. We can then find ⌈(45 − 18)/4⌉ = 7 copies of K 4 , with 17 vertices left over, in which we can find 2 monochromatic triangles, and distribute the remaining vertices so that each of these 11 parts has size 5. Arguing as above, this then implies f 5 (5n) ≥ 25f 5 (n) + 9(n − 9) + 37. (3.8) We omit the details, but using similar arguments and the Ramsey number bounds R(6, 6) ≤ 165 and R(7, 7) ≤ 540 yields the inequalities f 6 (6n) ≥ 36f 6 (n) + 14n − 151 (3.9) f 7 (7n) ≥ 49f 7 (n) + 20n − 532. (3.10) According to András Gyárfás [10], Paul Erdős, sitting in the Atlanta Airport in 1995, asked his companions whether every bicoloring of the edges of K R(k,k) contains two edge-disjoint monochromatic copies of K k . Ralph Faudree pointed out that this is not true, at which point Erdős asked for the smallest number n(k) for which any bicoloring of K n(k) does contain two edge-disjoint monochromatic K k 's. The next day, Faudree showed n(3) = 7, and some time later, Gyárfás showed n(4) = 19. For our purposes, however, we require vertex-disjoint monochromatic copies of K r . In the appendix we give an argument, inspired by the proof of n(4) = 19 by Gyárfás, showing that n = 20 is sufficient to find two vertex-disjoint monochromatic K 4 's, provided there is also a monochromatic K 5 : Lemma 3.7. Any bicoloring of the edges of K 20 with a monochromatic copy of K 5 contains two vertex-disjoint monochromatic copies of K 4 .
Proof. Consider any bicoloring of K 4n . Since 4n ≥ 48 ≥ R (5,5), there is some monochromatic copy of we consider the 4 n edge-colorings c of K n obtained by picking one vertex from each part. Each of these has a fractional clique packing ψ c of size at least f 4 (n), and since each edge is used in 4 −(n−2) such ψ c , we know that 4 −(n−2) c ψ c is a valid packing in K 4n . Adding the copies of K 4 and K 3 inside the n individual blocks, we see that f 4 (4n) ≥ 16f 4 (n) + 5(n − 3) + 6.

Computer-aided calculations
We next describe a generalization of the algorithm used by Keevash and Sudakov in the case of triangle packings [14], which we call the KS extension method. For any finite family of graphs F = {H 1 , . . . , H r }, any graph G ∈ G n , and any vector Γ ∈ R F , we let ν F ,Γ (G) be the value of the linear program n−1 · ℓ must be a one-vertex extension of some graph in L(G n , ℓ). In other words, if {ℓ n } n∈N is any sequence of numbers satisfying ℓ n+1 ≥ n+1 n−1 ℓ n , then L(G n+1 , ℓ n+1 ) ⊆ ext(L(G n , ℓ n )).
Let us refer to such sequences ℓ n as level sequences. Note that the sequence ℓ n used by Algorithm 1 does not have to be determined before runtime; as long as it is guaranteed to be a level sequence, this guarantees the loop invariant L(G n , ℓ n ) ⊆ L, and hence Λ[n] ≤ Λ F ,Γ (G n ).
In [14], they choose a parameter d (called the "search depth"), and define ℓ n recursively by taking ℓ n 0 = +∞ and ℓ n+1 to be n+1 n−1 · α n , where α n is either (a) the dth smallest value in the set {ν F ,Γ (G ′ ) + ν F ,Γ (G ′ ) : G ∈ L(G n , ℓ n )}, if this set has at least d elements, or (b) ℓ n , if the set has fewer than d elements. The role of d is to limit the number of graphs stored in the set L. If d = ∞, then Algorithm 1 has to solve the LP (3.11) on every graph up to size n in order to compute Λ F ,Γ (G n ), while if d is too small, then the while loop will terminate after a small number of iterations. We ran an implementation 3 of this method on a 24-core computing grid with d = 11, starting with an exhaustive search on n 0 = 6 vertices, and obtained the results summarized in Table 1. The last column in particular implies f 4 (20) > 64.725, which implies c 4 > 0.1703. Using Lemma 3.8, and inequalities (3.8), (3.9), and (3.10) (in that order), we can obtain the bound c 7 ≥ 0.1814, which implies max G∈Gn cp(G) + cp(G) < 0.3186n 2 + o(n 2 ).

Proof of Lemma 3.7
Lemma 3.7. Any bicoloring of the edges of K 20 with a monochromatic copy of K 5 contains two vertex-disjoint monochromatic copies of K 4 .
Proof. Suppose that we have a bicoloring of K 20 with a red copy N = {n 1 , ..., n 5 } of K 5 . If there is a blue copy of K 4 , then we are finished, because this blue copy and N cannot share an edge, and therefore share at most one vertex. We may now assume that all monochromatic copies of K 4 are red.
We can address the case in which there exists a vertex v such that it is incident to at least nine red and blue edges each relatively quickly. We denote by R and B the cliques on the red and blue neighbors of v, respectively. Because the Ramsey number R(3, 4) = 9 and our graph has no blue copy of K 4 , R must contain a red copy of K 3 . Moreover, B cannot contain a blue copy of K 3 , so B must contain a red copy of K 4 . Adding v to the red copy of K 3 in R results in two vertex-disjoint red copies of K 4 , one in R ∪ v and one in B. We may now assume that all vertices have at least eleven incident edges of the same color.
Consider the case in which some vertex v has two red and two blue edges adjacent to a red copy M of K 4 . If v has at least eleven red edges, then it has at least nine red edges connected to K 20 \(M ∪ {v}), which, by the same argument as above, implies K 20 \M has a red copy of K 4 . The same argument holds if v has at least eleven blue edges. We may now assume that no vertex has two red and two blue edges adjacent to a red copy of K 4 .
From here, we consider two cases: Case I: Suppose that there exists five vertices V = {v 1 , ..., v 5 } ⊂ K 20 \N, each with at least three red edges adjacent to N. Because no vertex has both two red and two blue edges adjacent to a red copy of K 4 , each vertex of V has at least four red edges adjacent to N. In addition, because our graph has no blue copy of K 4 every set V \v i has a red edge.
Suppose that some vertex of V , without loss of generality called v 1 , has five red edges adjacent to N. Without loss of generality, {v 2 , v 3 } is a red edge in V \v 1 . There are at most two blue edges from v 2 or v 3 to N; without loss of generality assume they are not incident to n 4 or n 5 . Then the subsets {v 1 , n 1 , n 2 , n 3 } and {v 2 , v 3 , n 4 , n 5 } are both red copies of K 4 . So we may now assume that each vertex in V has exactly four red edges adjacent to N.