Summary form only given. The edge connectivity of a graph G = (V, E) is defined as the function, /spl lambda/: V /spl times/ V /spl rarr/ N, that associates to any pair of vertices (u, v) the maximum number of edge-disjoint paths connecting the two vertices, /spl lambda/(u, v). In this paper, we present a method for determining the function /spl lambda/(u,v) for all vertex pairs in a 4-regular graph which achieves O(|V|) running time (with a small constant factor) and O(|V|) space complexity. We show with our method that determination and traversal of an Eulerian tour of each component of the 4-regular graph along with appropriate bookkeeping is enough for determining /spl lambda/(u, v) for all pairs (u,v).