Given a graph with a list of edges, is it possible to always construct a set of cycle bases for those edges, such that each and every edge is shared by at most 2 cycle bases?


The above question assumes that each and every edge must somehow belong to at least one cycle. IN other words, there is no vertex that is connected to one and only one edge. 



In a graph, is it always possible to construct a set of cycle bases, with each and every edge Is shared by at most 2 cycle bases?
Consider the complete graph on 7 vertices. It has 21 edges, so any set of cycles that utilizes each edge at most twice has size at most 42/3=14. But the cycle space of the graph has dimension 21-7+1=15, so you cannot have a basis with the requested property.


