Ordered Biclique Partition Number

• Ordered biclique partition number is an edge-decomposition parameter for complete graphs that requires each edge to be covered once or twice with a prescribed reversal in double coverings.
• It refines classical biclique partition theory by introducing ordering constraints, which reduce the asymptotic growth from linear in n to roughly n^(1/2+o(1)).
• Explicit combinatorial constructions and algebraic methods establish tight bounds for this parameter, impacting communication complexity and matrix theory.

to=arxiv_search.search 早点加盟 天天中彩票官方 ＿俺去也{"query":"Ordered biclique partition number complete graph Shigeta Amano 2013 2026 ordered biclique covering", "max_results": 10} to=arxiv_search.search {"query":"Ordered biclique partition number complete graph Shigeta Amano 2013 2026 ordered biclique covering", "max_results": 10} The ordered biclique partition number is an edge-decomposition parameter for the complete graph $K_n$ that refines both the ordinary biclique partition number and the $2$-biclique covering number. In the standard formulation, an ordered biclique partition of $K_n$ is a collection of bicliques such that every edge is covered at least once and at most twice, and whenever an edge is covered twice, its two endpoints appear in opposite bipartition classes across the two coverings. In the notation of almost balanced ordered biclique coverings, this parameter is $f(n,2)$; in earlier literature it is also denoted $\mathrm{bp}_{1.5}(K_n)$. Recent work places its asymptotic order at $\Theta(n^{1/2+o(1)})$ and frames it as a communication-complexity-motivated invariant (Babu et al., 7 Jun 2026).

1. Formal definition and notation

A biclique is a complete bipartite graph $B(U,W)$ on disjoint vertex sets $U,W$. For a graph $G$, the biclique partition number $\mathrm{bp}(G)$ is the minimum number of bicliques whose edge sets partition $2$0. More generally, a $2$1-biclique covering $2$2 allows each edge to be covered at least once and at most $2$3 times. The ordered biclique partition parameter adds an orientation constraint to the case $2$4: if an edge $2$5 is covered twice, then in one covering biclique $2$6 is in the first class and $2$7 in the second, while in the other covering biclique the roles are reversed (Shigeta et al., 2013).

The 2026 formulation places this in a broader family. An almost balanced ordered biclique covering of order $2$8 of $2$9 is a collection of bicliques such that:

1. each edge of $K_n$0 is contained at least once and at most $K_n$1 times;
2. for any edge $K_n$2 covered more than once, the number of times it is covered with $K_n$3 in the first part and $K_n$4 in the second differs by at most $K_n$5 from the number of times with $K_n$6 in the first part and $K_n$7 in the second;
3. the minimum number of bicliques in such a family is denoted $K_n$8.

For $K_n$9, this is precisely the ordered biclique partition number; for $f(n,2)$0, it reduces to the ordinary biclique partition number of $f(n,2)$1 (Babu et al., 7 Jun 2026).

2. Relation to classical biclique partition theory

The classical anchor point is the Graham–Pollak theorem, which states that

$f(n,2)$2

In the $f(n,2)$3 notation, this is the statement $f(n,2)$4. Thus the ordered biclique partition number is not a variant detached from classical theory, but rather the next case after exact edge partition into bicliques (Babu et al., 7 Jun 2026).

A useful comparison is with the unrestricted $f(n,2)$5-biclique covering number. Earlier work records

$f(n,2)$6

The lower bound comes from the trivial $f(n,2)$7-biclique covering number, while the upper bound reflects the additional ordered constraint. In this sense, the ordered parameter lies between exact biclique partitioning and general bounded-multiplicity covering: it allows overlap, but only under a prescribed reversal condition on double-covered edges (Shigeta et al., 2013).

This positioning explains why the parameter is structurally delicate. Exact biclique partitioning of $f(n,2)$8 is linear in $f(n,2)$9, whereas permitting controlled double coverage collapses the order of growth to roughly $\mathrm{bp}_{1.5}(K_n)$0. The ordered condition is therefore strong enough to matter combinatorially, but not strong enough to restore Graham–Pollak-type linear growth.

3. Asymptotic bounds for $\mathrm{bp}_{1.5}(K_n)$1

For the ordered biclique partition number of the complete graph, the previously best known bounds were

$\mathrm{bp}_{1.5}(K_n)$2

for some positive constants $\mathrm{bp}_{1.5}(K_n)$3. The 2026 work establishes almost tight bounds for $\mathrm{bp}_{1.5}(K_n)$4 for general $\mathrm{bp}_{1.5}(K_n)$5, and in particular sharpens the formulation for $\mathrm{bp}_{1.5}(K_n)$6 (Babu et al., 7 Jun 2026).

Its lower-bound theorem states that for $\mathrm{bp}_{1.5}(K_n)$7,

$\mathrm{bp}_{1.5}(K_n)$8

Specializing to $\mathrm{bp}_{1.5}(K_n)$9 yields

$\Theta(n^{1/2+o(1)})$0

for some constant $\Theta(n^{1/2+o(1)})$1.

Its upper-bound theorem states that for $\Theta(n^{1/2+o(1)})$2,

$\Theta(n^{1/2+o(1)})$3

For $\Theta(n^{1/2+o(1)})$4, this becomes

$\Theta(n^{1/2+o(1)})$5

for some constant $\Theta(n^{1/2+o(1)})$6.

Taken together, these results confirm

$\Theta(n^{1/2+o(1)})$7

The paper explicitly states that this matches the lower and upper bounds up to lower-order terms and constants, aligns with the previous best-known asymptotic bounds, and provides improved clarity and construction details for the ordered biclique partition number (Babu et al., 7 Jun 2026).

4. Constructions and proof techniques

The first explicit near-optimal upper bound was given by Shigeta and Amano. Their construction improved the earlier $\Theta(n^{1/2+o(1)})$8 bound to

$\Theta(n^{1/2+o(1)})$9

The construction labels the vertices of a complete graph by tuples

$B(U,W)$0

defines several classes of edge sets denoted $B(U,W)$1, $B(U,W)$2, and $B(U,W)$3, and partitions the induced subgraphs by stars. A key feature is that the ordered covering property is enforced by lexicographic and reverse orderings on coordinates when choosing star roots, so that when an edge is covered twice the two orientations are complementary (Shigeta et al., 2013).

The same paper includes a concrete ordered biclique partition of $B(U,W)$4: $B(U,W)$5 This example is used as an explicit illustration that the ordered condition can be met with overlap, rather than by exact partition alone.

The 2026 lower bound uses a polynomial method: each vertex is assigned a multivariate polynomial, the resulting family of polynomials is shown to be linearly independent, and the dimension of the ambient polynomial space bounds the size of the biclique family from below. The corresponding upper bound is constructive and uses binary vectors with Hamming weight constraints, a hypercube-related encoding, known constructions of ordered biclique partitions from Shigeta and Amano, and a product construction to maximize family size while preserving the balance and covering constraints (Babu et al., 7 Jun 2026).

These two strands—explicit combinatorial constructions and algebraic lower bounds—now constitute the standard proof architecture around the parameter.

5. Connections to communication complexity and matrix theory

The ordered biclique partition number is closely tied to communication complexity. The 2013 construction was developed in part to address questions on fooling sets, Boolean matrix rank, and the clique-vs.-independent-set problem (Shigeta et al., 2013).

One consequence is the construction of $B(U,W)$6 $B(U,W)$7-matrices of rank $B(U,W)$8 that have a fooling set of size $B(U,W)$9. Equivalently, the paper establishes an almost quadratic gap,

$U,W$0

The mechanism is to form rank-$U,W$1 matrices from the bicliques in an ordered biclique partition and sum them to obtain a low-rank matrix with a large fooling set.

A second consequence is an improved lower bound

$U,W$2

on the nondeterministic communication complexity of the clique vs. independent set problem. The paper states that this matches the best known lower bound on the deterministic version of the problem.

These consequences are not incidental. They explain why the parameter is described as communication-complexity-motivated: the edge-orientation constraint in double coverings is exactly the feature that allows graph decompositions to transfer into rectangle-based and rank-based lower-bound constructions.

6. Related notions and distinctions

Several nearby parameters can be confused with the ordered biclique partition number, but the literature distinguishes them sharply.

First, the ordinary biclique partition number on other graph classes admits exact or nearly exact structural formulas. For split graphs,

$U,W$3

where $U,W$4 is the number of maximal cliques in the complement; this extends the Graham–Pollak theorem from complete graphs to all split graphs (Babu et al., 10 Jul 2025). For co-chordal graphs, constructive heuristics yield

$U,W$5

and if $U,W$6 is chordal and clique vertex irreducible, then equality holds (Lyu et al., 2022). These are results about exact edge partition, not ordered double coverage.

Second, in work on biclique decompositions from Welzl orders, “ordered” refers to a vertex ordering under which neighborhoods become unions of few intervals. There the size of a decomposition is measured as the sum of the numbers of vertices of its bicliques, not the number of bicliques. The paper proves that if a graph $U,W$7 on $U,W$8 vertices has contiguity $U,W$9, then $G$0 has a biclique decomposition with at most $G$1 bicliques, every vertex appears in at most $G$2 bicliques, and hence

$G$3

This is a different optimization problem, despite the shared emphasis on order (Cardinal et al., 8 Jun 2026).

Third, the biclique vertex-partition number arising from poset theory is again distinct. For the bipartite transformation $G$4 of a poset $G$5, one has

$G$6

for $G$7, where the right-hand side is a minimum partition of the vertex set into bicliques, not a minimum partition or almost balanced covering of the edge set (Civan et al., 2020).

The terminology “ordered biclique partition number” is therefore specific: it refers to the minimum number of bicliques in an edge-covering family for $G$8 with multiplicity at most $G$9 and a prescribed reversal rule on double-covered edges.

7. Open direction and present status

The principal asymptotic question for $\mathrm{bp}(G)$0 is largely settled at the level of exponents: the ordered biclique partition number has growth $\mathrm{bp}(G)$1 (Babu et al., 7 Jun 2026). Earlier work already asked whether the $\mathrm{bp}(G)$2 term in the exponent can be removed and whether one can achieve

$\mathrm{bp}(G)$3

without subpolynomial slack (Shigeta et al., 2013).

The available evidence is two-sided. On the one hand, explicit constructions achieve $\mathrm{bp}(G)$4, and modern generalizations produce constructive upper bounds through binary-vector and product constructions. On the other hand, polynomial-independence arguments now supply matching lower bounds up to lower-order terms. The present status is therefore asymptotically stable but not fully constant-sharp.

Within combinatorics and theoretical computer science, the ordered biclique partition number occupies a specific niche: it is a complete-graph decomposition invariant interpolating between Graham–Pollak exact partitioning and bounded-multiplicity biclique covering, while simultaneously encoding communication-complexity phenomena through its orientation-sensitive overlap structure.