Graphs and Combinatorics

, Volume 35, Issue 4, pp 921–931

# Berge’s Conjecture and Aharoni–Hartman–Hoffman’s Conjecture for Locally In-Semicomplete Digraphs

Original Paper

## Abstract

Let k be a positive integer and let D be a digraph. A path partition $$\mathcal {P}$$ of D is a set of vertex-disjoint paths which covers V(D). Its k-norm is defined as $$\sum _{P \in \mathcal {P}} \min \{|V(P)|, k\}$$. A path partition is k-optimal if its k-norm is minimum among all path partitions of D. A partialk-coloring is a collection of k disjoint stable sets. A partial k-coloring $$\mathcal {C}$$ is orthogonal to a path partition $$\mathcal {P}$$ if each path $$P \in \mathcal {P}$$ meets $$\min \{|V(P)|,k\}$$ distinct sets of $$\mathcal {C}$$. Berge (Eur J Comb 3(2):97–101, 1982) conjectured that every k-optimal path partition of D has a partial k-coloring orthogonal to it. A (path) k-pack of D is a collection of at most k vertex-disjoint paths in D. Its weight is the number of vertices it covers. A k-pack is optimal if its weight is maximum among all k-packs of D. A coloring of D is a partition of V(D) into stable sets. A k-pack $$\mathcal {P}$$ is orthogonal to a coloring $$\mathcal {C}$$ if each set $$C \in \mathcal {C}$$ meets $$\min \{|C|, k\}$$ paths of $$\mathcal {P}$$. Aharoni et al. (Pac J Math 2(118):249–259, 1985) conjectured that every optimal k-pack of D has a coloring orthogonal to it. A digraph D is semicomplete if every pair of distinct vertices of D are adjacent. A digraph D is locally in-semicomplete if, for every vertex $$v \in V(D)$$, the in-neighborhood of v induces a semicomplete digraph. Locally out-semicomplete digraphs are defined similarly. In this paper, we prove Berge’s and Aharoni–Hartman–Hoffman’s Conjectures for locally in/out-semicomplete digraphs.

## Keywords

Path partition Coloring Berge’s Conjecture Aharoni, Hartman, and Hoffman’s Conjecture Locally in-semicomplete digraphs

## References

1. 1.
Aharoni, R., Hartman, I.B.A.: On Greene–Kleitman’s theorem for general digraphs. Discrete Math. 120(1–3), 13–24 (1993)
2. 2.
Aharoni, R., Hartman, I.B.A., Hoffman, A.J.: Path partitions and packs of acyclic digraphs. Pac. J. Math. 2(118), 249–259 (1985)
3. 3.
Bang-Jensen, J.: Locally semicomplete digraphs: A generalization of tournaments. J. Graph Theory 14(3), 371–390 (1990)
4. 4.
Bang-Jensen, J.: Digraphs with the path-merging property. J. Graph Theory 20(2), 255–265 (1995)
5. 5.
Bang-Jensen, J., Guo, Y., Gutin, G., Volkmann, L.: A classification of locally semicomplete digraphs. Discrete Math. 167, 101–114 (1997)
6. 6.
Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications, 2nd edn. Springer Monographs in Mathematics, Springer, London (2009)
7. 7.
Bang-Jensen, J., Nielsen, M.H., Yeo, A.: Longest path partitions in generalizations of tournaments. Discrete Math. 306(16), 1830–1839 (2006)
8. 8.
Berge, C.: $$k$$-optimal partitions of a directed graph. Eur. J. Comb. 3(2), 97–101 (1982)
9. 9.
Berger, E., Hartman, I.B.A.: Proof of Berge’s strong path partition conjecture for $$k = 2$$. Eur. J. Comb. 29(1), 179–192 (2008)
10. 10.
Cameron, K.: On $$k$$-optimum dipath partitions and partial $$k$$-colourings of acyclic digraphs. Eur. J. Comb. 7(2), 115–118 (1986)
11. 11.
Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51(1), 161–166 (1950)
12. 12.
Galeana-Sánchez, H., Olsen, M.: A characterization of locally semicomplete CKI-digraphs. Graphs Comb. 32(5), 1873–1879 (2016)
13. 13.
Galeana-Sánchez, H., Gómez, R.: Independent sets and non-augmentable paths in generalizations of tournaments. Discrete Math. 308(12), 2460–2472 (2008)
14. 14.
Gallai, T.: On directed paths and circuits. Theory Graphs 38, 115–118 (1968)
15. 15.
Gallai, T., Milgram, A.N.: Verallgemeinerung eines graphentheoretischen Satzes von Rédei. Acta Sci. Math. 21, 181–186 (1960)
16. 16.
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
17. 17.
Greene, C.: Some partitions associated with a partially ordered set. J. Comb. Theory Ser. A 20(1), 69–79 (1976)
18. 18.
Greene, C., Kleitman, D.J.: The structure of Sperner $$k$$-families. J. Comb. Theory Ser. A 20(1), 41–68 (1976)
19. 19.
Guo, Y., Volkmann, L.: Connectivity properties of locally semicomplete digraphs. J. Graph Theory 18(3), 269–280 (1994a)Google Scholar
20. 20.
Guo, Y., Volkmann, L.: On complementary cycles in locally semicomplete digraphs. Discrete Math. 135(1), 121–127 (1994b)Google Scholar
21. 21.
Hartman, I.B.A., Saleh, F., Hershkowitz, D.: On Greene’s theorem for digraphs. J. Graph Theory 18(2), 169–175 (1994)
22. 22.
Herskovics, D.: Proof of Berge’s path partition conjecture for $$k \ge \lambda -3$$. Discrete Appl. Math. 209, 137–143 (2016) (9th International Colloquium on Graph Theory and Combinatorics, 2014, Grenoble)Google Scholar
23. 23.
Huang, J.: A note on spanning local tournaments in locally semicomplete digraphs. Discrete Appl. Math. 89(1), 277–279 (1998)
24. 24.
Linial, N.: Covering digraphs by paths. Discrete Math. 23(3), 257–272 (1978)
25. 25.
Linial, N.: Extending the Greene–Kleitman theorem to directed graphs. J. Comb. Theory Ser. A 30(3), 331–334 (1981)
26. 26.
Mirsky, L.: A dual of Dilworth’s decomposition theorem. Am. Math. Mon. 78, 876–877 (1971)
27. 27.
Roy, B.: Nombre chromatique et plus longs chemins d’un graphe. Rev. Française d’informatique et de Recherche Opérationnelle 1(5), 129–132 (1967)
28. 28.
Sridharan, S.: On the strong path partition conjecture of Berge. Discrete Math. 117(1–3), 265–270 (1993)

© Springer Japan KK, part of Springer Nature 2019

## Authors and Affiliations

• Maycon Sambinelli
• 1
Email author
• Carla Negri Lintzmayer
• 2
• Cândida Nunes da Silva
• 3
• Orlando Lee
• 1
1. 1.Institute of ComputingUniversity of CampinasCampinasBrazil
2. 2.Center for MathematicsComputing and Cognition, Federal University of ABC, Santo AndréSão PauloBrazil
3. 3.Department of ComputingFederal University of São CarlosSorocabaBrazil

## Personalised recommendations

### Citearticle 