Graphs and Combinatorics

, Volume 35, Issue 4, pp 921–931 | Cite as

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

  • Maycon SambinelliEmail author
  • Carla Negri LintzmayerEmail author
  • Cândida Nunes da Silva
  • Orlando Lee
Original Paper


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.


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



M. Sambinelli was supported by National Counsel of Technological and Scientific Development—CNPq (Proc. 141216/2016-6), C. N. Lintzmayer by São Paulo Research Foundation—FAPESP (Proc. 2016/14132-4), and O. Lee by CNPq (Proc. 311373/2015-1) and FAPESP (Proc. 2015/11937-9).


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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  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

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