Advertisement

Combinatorica

, Volume 39, Issue 2, pp 239–263

# Edge-Partitioning a Graph into Paths: Beyond the Barát-Thomassen Conjecture

• Julien Bensmail
• Ararat Harutyunyan
• Tien-Nam Le
• Stéphan Thomassé
Article
• 52 Downloads

## 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.

## Mathematics Subject Classification (2010)

05C40 05C07 05C15 05C38

## Preview

Unable to display preview. Download preview PDF.

## References

1. [1]
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.
2. [2]
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.
3. [3]
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.Google Scholar
4. [4]
J. Barát and C. Thomassen: Claw-decompositions and Tutte-orientations. Journal of Graph Theory 52 (2006), 135–146.
5. [5]
J. Edmonds: Edge-disjoint branchings, Combinatorial Algorithms (B. Rustin, editor), 91–96, Academic Press, 1973.Google Scholar
6. [6]
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.
7. [7]
C. McDiarmid: Concentration for Independent Permutations. Combinatorics, Probability and Computing 11 (2002), 163–178.
8. [8]
M. Molloy and B. Reed: Graph Colouring and the Probabilistic Method. Springer, 2002.
9. [9]
C. St. J. A. Nash-Williams: On orientations, connectivity and odd-vertex-pairings in finite graphs. Canadian Journal of Mathematics 12 (1960), 555–567.
10. [10]
M. Stiebitz, D. Scheide, B. Toft and L. M. Favrholdt: Graph Edge Coloring: Vizing's Theorem and Goldberg's Conjecture, Wiley, 2012.
11. [11]
C. Thomassen: Decompositions of highly connected graphs into paths of length 3. Journal of Graph Theory 58 (2008), 286–292.
12. [12]
C. Thomassen: Edge-decompositions of highly connected graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 18 (2008), 17–26.
13. [13]
C. Thomassen: Decomposing graphs into paths of fixed length. Combinatorica 33 (2013), 97–123.

## Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2018

## Authors and Affiliations

• Julien Bensmail
• 1
Email author
• Ararat Harutyunyan
• 2
• Tien-Nam Le
• 3
• Stéphan Thomassé
• 3
1. 1.I3S and INRIAUniversité Nice-Sophia-AntipolisSophia-AntipolisFrance
2. 2.LAMSADE, CNRSUniversité Paris-Dauphine PSL Research UniversityParisFrance
3. 3.Laboratoire d’Informatique du ParallélismeÉcole Normale Supérieure de LyonLyon Cedex 07France