Abstract
In 2006, Barát and Thomassen conjectured that there is a function f such that, for every fixed tree T with t edges, every f(t)-edge-connected graph with its number of edges divisible by t has a partition of its edges into copies of T. This conjecture was recently verified by the current authors and Merker [1].
We here further focus on the path case of the Barát-Thomassen conjecture. Before the aforementioned general proof was announced, several successive steps towards the path case of the conjecture were made, notably by Thomassen [11,12,13], until this particular case was totally solved by Botler, Mota, Oshiro andWakabayashi [2]. Our goal in this paper is to propose an alternative proof of the path case with a weaker hypothesis: Namely, we prove that there is a function f such that every 24-edge-connected graph with minimum degree f(t) has an edge-partition into paths of length t whenever t divides the number of edges. We also show that 24 can be dropped to 4 when the graph is eulerian.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Bensmail, A. Harutyunyan, T.-N. Le, M. Merker and S. Thomassé: A Proof of the Barát-Thomassen Conjecture. Journal of Combinatorial Theory, Series B 124 (2017), 39–55.
F. Botler, G. O. Mota, M. Oshiro and Y. Wakabayashi: Decomposing highly edge-connected graphs into paths of any given length. Journal of Combinatorial Theory, Series B 122 (2017), 508–542.
F. Botler, G. O. Mota, M. Oshiro and Y. Wakabayashi: Decompositions of highly connected graphs into paths of length five. Discrete Applied Mathematics, Doi: 10.1016/j.dam.2016.08.001, 2016.
J. Barát and C. Thomassen: Claw-decompositions and Tutte-orientations. Journal of Graph Theory 52 (2006), 135–146.
J. Edmonds: Edge-disjoint branchings, Combinatorial Algorithms (B. Rustin, editor), 91–96, Academic Press, 1973.
B. Jackson: On circuit covers, circuit decompositions and Euler tours of graph, Surveys in Combinatorics, London Mathematical Society Lecture Note Series, 187 (1993), 191–210.
C. McDiarmid: Concentration for Independent Permutations. Combinatorics, Probability and Computing 11 (2002), 163–178.
M. Molloy and B. Reed: Graph Colouring and the Probabilistic Method. Springer, 2002.
C. St. J. A. Nash-Williams: On orientations, connectivity and odd-vertex-pairings in finite graphs. Canadian Journal of Mathematics 12 (1960), 555–567.
M. Stiebitz, D. Scheide, B. Toft and L. M. Favrholdt: Graph Edge Coloring: Vizing's Theorem and Goldberg's Conjecture, Wiley, 2012.
C. Thomassen: Decompositions of highly connected graphs into paths of length 3. Journal of Graph Theory 58 (2008), 286–292.
C. Thomassen: Edge-decompositions of highly connected graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 18 (2008), 17–26.
C. Thomassen: Decomposing graphs into paths of fixed length. Combinatorica 33 (2013), 97–123.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported by ERC Advanced Grant GRACOL, project no. 320812. The second author was supported by an FQRNT postdoctoral research grant and CIMI research fellowship. The fourth author was partially supported by the ANR Project STINT under Contract ANR-13-BS02-0007.
Rights and permissions
About this article
Cite this article
Bensmail, J., Harutyunyan, A., Le, TN. et al. Edge-Partitioning a Graph into Paths: Beyond the Barát-Thomassen Conjecture. Combinatorica 39, 239–263 (2019). https://doi.org/10.1007/s00493-017-3661-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-017-3661-5