Connectivity graph‐codes

Abstract

AbstractThe symmetric difference of two graphs on the same set of vertices is the graph on whose set of edges are all edges that belong to exactly one of the two graphs . For a fixed graph call a collection of spanning subgraphs of a connectivity code for if the symmetric difference of any two distinct subgraphs in is a connected spanning subgraph of . It is easy to see that the maximum possible cardinality of such a collection is at most , where is the edge‐connectivity of and is its minimum degree. We show that equality holds for any ‐regular (mild) expander, and observe that equality does not hold in several natural examples including any large cubic graph, the square of a long cycle and products of a small clique with a long cycle.