Countable locally 2-arc-transitive bipartite graphs

We present an order-theoretic approach to the study of countably infinite locally 2-arc-transitive bipartite graphs. Our approach is motivated by techniques developed by Warren and others during the study of cycle-free partial orders. We give several new families of previously unknown countably infinite locally-2-arc-transitive graphs, each family containing continuum many members. These examples are obtained by gluing together copies of incidence graphs of semilinear spaces, satisfying a certain symmetry property, in a tree-like way. In one case we show how the classification problem for that family relates to the problem of determining a certain family of highly arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page

Short Cycles Connectivity

Short cycles connectivity is a generalization of ordinary connectivity. Instead by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two adjacent cycles have at least one common vertex. If all adjacent cycles in the sequence share at least one edge, we talk about edge short cycles connectivity. It is shown that the short cycles connectivity is an equivalence relation on the set of vertices, while the edge short cycles connectivity components determine an equivalence relation on the set of edges. Efficient algorithms for determining equivalence classes are presented. Short cycles connectivity can be extended to directed graphs (cyclic and transitive connectivity). For further generalization we can also consider connectivity by small cliques or other families of graphs