Notes about math, research, and more.
Questions
- Do all coarse disjoint union of a finite sequence of graphs of
growing girth which coarsely embed into sequence of cubes “look
like” some \(Z_{m}\)-homology cover?
- The thought is that a hypercube is made from two smaller hypercubes which are made from two smaller hypercubes… If the girth of the graph is larger than \(2^k\) then none of the \(k\)-dimensional hypercubes can contain a cycle. So a graph with large girth inside a hypercube is going to be trees inside the smaller cubes, the edges which create the cycles are between the smaller cubes which is what a \(Z_{m}\)-homology cover is (though in the homology cover all the trees are the same, but coarsely how different can two trees which are in the same hypercubes be?)
- How does property A relate to sequence of cubes?
- Can we modify a construction for girth with large graphs so that it is confined to subgraphs of hypercubes? Biggs does not work directly, there are too few vertices to choose from (for each \(x\) there is only a single \(y\) that is distance \(log_{2}(V(X_{n}))\) from \(x\)).