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Heinrich et al. [4] characterized those simple eulerian graphs with no Petersen-minor which admit a triangle-free cycle decomposition, a TFCD. If one permits Petersen minors then no such characterization is known even for \( {E}(4,2) \), the set of all the eulerian graphs of maximum degree 4. Let \( {EM}(4,2) \subset {E}(4,2) \) be the set of all graphs \( H \) such that all triangles of \( H \) are vertex disjoint, and each triangle contains a degree 2 vertex in \( H \). In the paper it is shown that to each \( G \in {E}(4,2) \) there exists a finite subset \( S \subset {EM}(4,2) \) so that \( G \) admits a TFCD if and only if some \( H \in S \) admits a TFCD. Further, some sufficient conditions for a graph \( G \in {E}(4,2) \) to possess a TFCD are given.