Eigensharp Graphs: Decomposition into Complete Bipartite Subgraphs

On 2-colored graphs and partitions of boxes

“…It may happen that this change causes a violation of (4), namely when we had R j * \ j ′ =j * R j ′ = {v} for some j * = j before the change; in this case, after adding v to R j we discard R j * . Iterating this operation we end up with families satisfying (1)- (5).…”

“…For more general results about decomposition of an arbitrary graph G, see e.g. Kratzke et al [5]. The case when G itself is complete bipartite is of course uninteresting, because there is a decomposition into one subgraph.…”

On the decomposition of random hypergraphs

“…Let α(G) be the independence number of G. It is easy to observe τ (G) ≤ |V (G)| − α(G). Erdős (see [13]) conjectured that the equality holds for almost all graphs. Namely, if G ∈ G(n, 1/2), then τ (G) = n − α(G) with high probability.…”

Addressing the Petersen graph

“…where D is the n×n matrix with distance entries d i,j =d(i, j ), and h(D)=max{n + (D), n − (D)} is the maximum of the numbers of positive and negative eigenvalues of D. Using a term employed by Kratzke et al [7], an addressing of G of length N will be called eigensharp if N = h(D). Since N N(G) h(D), eigensharp addressings are optimal.…”

“…It is known [8, p. 62] that trees, complete graphs and odd cycles have eigensharp addressings of length n − 1, while even cycles have eigensharp addressings of length n/2. If G has a length k addressing using only the two symbols a, b, equivalently, if G may be isometrically embedded into the k-cube, then results of Graham and Winkler [4,Theorems 3,7] imply that such an addressing may be chosen to have length n − (D) and so be eigensharp. In particular, the length k addressing of the k-cube obtained by replacing the 1's and 0's of its vertex k-tuples by a's and b's is eigensharp.…”