Geodesic
On path-cycle decompositions of triangle-free graphs
Discrete mathematics & theoretical computer science, Tome 19 (2017-2018) no. 3.

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In this work, we study conditions for the existence of length-constrained path-cycle decompositions, that is, partitions of the edge set of a graph into paths and cycles of a given minimum length. Our main contribution is the characterization of the class of all triangle-free graphs with odd distance at least $3$ that admit a path-cycle decomposition with elements of length at least $4$. As a consequence, it follows that Gallai's conjecture on path decomposition holds in a broad class of sparse graphs. 
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     author = {Jim\'enez, Andrea and Wakabayashi, Yoshiko},
     title = {On path-cycle decompositions of triangle-free graphs},
     journal = {Discrete mathematics & theoretical computer science},
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Jiménez, Andrea; Wakabayashi, Yoshiko. On path-cycle decompositions of triangle-free graphs. Discrete mathematics & theoretical computer science, Tome 19 (2017-2018) no. 3. doi : 10.23638/DMTCS-19-3-7. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-19-3-7/

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