We will denote an Euler tour in G by the sequence of vertices visited by the tour.

9] proved that the graphs in G are the only connected (2,4)-graphs that do not admit a triangle-free Euler tour.

Theorem 10 [9] A connected (2,4)-graph G admits a triangle-free Euler tour if and only if G [?

4]-decomposition of G by properly segmenting a triangle-free Euler tour in G.

3])-free Euler tour E* and we may assume without loss of generality that u, w, v is a subsequence of E*.

9] proved the following theorem as a corollary of a theorem on triangle-free Euler tours in connected 4-regular graphs.

Given a rooted tree T' = (V, E), the Euler Tour ET(T') of T' is a sequence of nodes.

Construct the Euler Tour ET(T) or compute the depth of each node.

T] can be computed from T in O(Sort(m)) I/Os by exploiting the Euler Tour ET(T) (Theorem 4.

Then we advance in each Euler-Tour list as long as the characters labeling the two examined edges match; otherwise, output to the disk the Euler tour of the subtree with the path from the root labeled by the lexicographically smaller string.

Our solution follows this basic idea but takes advantage of the Euler Tours of the two uncompacted tries as follows.