Infinite Euler Graphs

It is well known that for finite connected graphs the following are equivalent: (i) X is Euler (i.e., every vertex of X has positive even degree); (ii) X is traceable (i.e., the edges of X can be arranged in a sequence e 1, . . . ,en such that ei ≠ ej if i ≠ j, and ei , e i+1 are adjacent, i = 1, . . . , n, subscripts considered mod n) ; (iii) X is cyclically coverable (i.e., X contains a family of non-overlapping circuits whose union is X).