On the spanning tree packing number of a graph: a survey

AbstractThe spanning tree packing number or STP number of a graph G is the maximum number of edge-disjoint spanning trees contained in G. We use an observation of Paul Catlin to investigate the STP numbers of several families of graphs including quasi-random graphs, regular graphs, complete bipartite graphs, cartesian products and the hypercubes

A Simple Algorithm for Graph Reconstruction

How efficiently can we find an unknown graph using distance queries between its vertices? We assume that the unknown graph is connected, unweighted, and has bounded degree. The goal is to find every edge in the graph. This problem admits a reconstruction algorithm based on multi-phase Voronoi-cell decomposition and using $\tilde O(n^{3/2})$ distance queries. In our work, we analyze a simple reconstruction algorithm. We show that, on random $\Delta$-regular graphs, our algorithm uses $\tilde O(n)$ distance queries. As by-products, we can reconstruct those graphs using $O(\log^2 n)$ queries to an all-distances oracle or $\tilde O(n)$ queries to a betweenness oracle, and we bound the metric dimension of those graphs by $\log^2 n$. Our reconstruction algorithm has a very simple structure, and is highly parallelizable. On general graphs of bounded degree, our reconstruction algorithm has subquadratic query complexity

Connectivity Graph-Codes

The symmetric difference of two graphs $G_1,G_2$ on the same set of vertices $V$ is the graph on $V$ whose set of edges are all edges that belong to exactly one of the two graphs $G_1,G_2$. For a fixed graph $H$ call a collection ${\cal G}$ of spanning subgraphs of $H$ a connectivity code for $H$ if the symmetric difference of any two distinct subgraphs in ${\cal G}$ is a connected spanning subgraph of $H$. It is easy to see that the maximum possible cardinality of such a collection is at most $2^{k'(H)} \leq 2^{\delta(H)}$, where $k'(H)$ is the edge-connectivity of $H$ and $\delta(H)$ is its minimum degree. We show that equality holds for any $d$-regular (mild) expander, and observe that equality does not hold in several natural examples including powers of long cycles and products of a small clique with a long cycle