Abstract
Let T be a tree with m edges. It was conjectured that every m-regular bipartite graph can be decomposed into edge-disjoint copies of T. In this paper, we prove that every 6-regular bipartite graph can be decomposed into edge-disjoint paths with 6 edges. As a consequence, every 6-regular bipartite graph on n vertices can be decomposed into \(\frac{n}{2}\) paths, which is related to the well-known Gallai’s Conjecture: every connected graph on n vertices can be decomposed into at most \(\frac{n+1}{2}\) paths.
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Chu, Y., Fan, G. & Zhou, C. Decompositions of 6-Regular Bipartite Graphs into Paths of Length Six. Graphs and Combinatorics 37, 263–269 (2021). https://doi.org/10.1007/s00373-020-02251-z
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DOI: https://doi.org/10.1007/s00373-020-02251-z