We say $G$ is \emph{$(Q_n,Q_m)$-saturated} if it is a maximal $Q_m$-free subgraph of the $n$-dimensional hypercube $Q_n$. A graph, $G$, is said to be $(Q_n,Q_m)$-semi-saturated if it is a subgraph of $Q_n$ and adding any edge forms a new copy of $Q_m$. The minimum number of edges a $(Q_n,Q_m)$-saturated graph (resp. $(Q_n,Q_m)$-semi-saturated graph) can have is denoted by $sat(Q_n,Q_m)$ (resp. $s\text{-}sat(Q_n,Q_m)$). We prove that $ \lim_{n\to\infty}\frac{sat(Q_n,Q_m)}{e(Q_n)}=0$, for fixed $m$, disproving a conjecture of Santolupo that, when $m=2$, this limit is $\frac{1}{4}$. Further, we show by a different method that $sat(Q_n, Q_2)=O(2^n)$, and that $s\text{-}sat(Q_n, Q_m)=O(2^n)$, for fixed $m$. We also prove the lower bound $s-sat(Q_n,Q_2)\geq \frac{m+1}{2}\cdot 2^n$, thus determining $sat(Q_n,Q_2)$ to within a constant factor, and discuss some further questions.Comment: Journal version, 16 pages, 1 figur
The biclique cover number (resp. biclique partition number ) of a graph G, bc(G) (resp. bp(G)), is the least number of bicliquescomplete bipartite subgraphs-that are needed to cover (resp. partition) the edges of G.The local biclique cover number (resp. local biclique partition number ) of a graph G, lbc(G) (resp. lbp(G)), is the least r such that there is a cover (resp. partition) of the edges of G by bicliques with no vertex in more than r of these bicliques.We show that bp(G) may be bounded in terms of bc(G), in particular, bp(G) ≤ 1 2 (3 bc(G) − 1). However, the analogous result does not hold for the local measures. Indeed, in our main result, we show that lbp(G) can be arbitrarily large, even for graphs with lbc(G) = 2. For such graphs, G, we try to bound lbp(G) in terms of additional information about biclique covers of G. We both answer and leave open questions related to this.There is a well known link between biclique covers and subcube intersection graphs. We consider the problem of finding the least r(n) for which every graph on n vertices can be represented as a subcube intersection graph in which every subcube has dimension r. We reduce this problem to the much studied question of finding the least d(n) such that every graph on n vertices is the intersection graph of subcubes of a d-dimensional cube.
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Let A and B be disjoint sets, of size 2 k , of vertices of Qn, the ndimensional hypercube. In 1997, Bollobás and Leader proved that there must be (n − k)2 k edge-disjoint paths between such A and B. They conjectured that when A is a down-set and B is an up-set, these paths may be chosen to be directed (that is, the vertices in the path form a chain). We use a novel type of compression argument to prove stronger versions of these conjectures, namely that the largest number of edgedisjoint paths between a down-set A and an up-set B is the same as the largest number of directed edge-disjoint paths between A and B. Bollobás and Leader made an analogous conjecture for vertex-disjoint paths and we prove a strengthening of this by similar methods. We also prove similar results for all other sizes of A and B. * Supported by an EPSRC doctoral studentship. Edge disjoint paths in the cubeThe edge boundary of a subset S of P[n], is written ∂ e (S), the set of Q n -edges with exactly one endpoint in S. The directed edge boundary, written − → ∂ e (S), is the set of edges in ∂ e (S) with smaller endpoint in S.The Edge Isoperimetric Inequality answers the extremal problem of which sets, of a given size, have smallest edge boundary. To state the theorem, we must define the binary order : we let x < y if max(x△y) ∈ y. Thus for all k, the subcube P[k] is an initial segment of the binary order on P[n]. The Edge Isoperimetric Inequality, proved by Harper [6], Lindsey [9], Bernstein [3] and Hart [8] states that initial segments minimize the size of the edge boundary.Theorem 1 (Edge Isoperimetric Inequality). Let A ⊆ P[n]. Let I be the set of the first |A| elements of P[n] in the binary order. Then |∂ e (A)| ≥ |∂ e (I)|. In particular, if |A| = 2 k , then its edge boundary is larger than that of a kdimensional subcube; i.e. |∂ e (A)| ≥ (n − k)2 n .We write p e (A, B) for the size of the largest collection of edge-disjoint paths between two disjoint subsets of the cube, A and B. Similarly, we write − → p e (A, B) for the size of the largest collection of edge-disjoint directed paths between disjoint A and B. In 1997, Bollobás and Leader [2], gave a lower bound on p e (A, B), in terms of |A| and |B|.Theorem 2 (Bollobás-Leader [2]). Let A and B be disjoint subsets of Q n , each of size 2 k , for some non-negative integer k. Then there is a family of at least (n − k)2 k edge-disjoint directed paths from A to B.
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