Abstract
It is known that the edge set of a 2-edge-connected 3-regular graph can be decomposed into paths of length 3. W. Li asked whether the edge set of every 2-edge-connected graph can be decomposed into paths of length at least 3. The graphs C 3, C 4, C 5, and K 4−e have no such decompositions. We construct an infinite sequence {F i } ∞ i=0 of nondecomposable graphs. On the other hand, we prove that every other 2-edge-connected graph has a desired decomposition.
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References
Li, W.: Problem 9.26, In: I. Rival (ed.), Graphs and Order, D. Reidel Publishing Co., Dordrecht, 1985.
Petersen, J.: Die Theorie der regulären Graphen, Acta Math. 15 (1891), 193–220.
West, D. B.: Introduction to Graph Theory, 2nd edn, Prentice-Hall, Upper Saddle River, 2001, Problem 3.3.20 on p. 147.
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Kostochka, A., Tashkinov, V. Decomposing Graphs into Long Paths. Order 20, 239–253 (2003). https://doi.org/10.1023/B:ORDE.0000026488.11602.7b
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DOI: https://doi.org/10.1023/B:ORDE.0000026488.11602.7b