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

Author(s)

Sambinelli, Maycon ^[1] ; Lintzmayer, Carla Negri ^[2] ; da Silva, Candida Nunes ^[3] ; Lee, Orlando ^[1]

Affiliation

^[1] Univ Estadual Campinas, Inst Comp, Campinas, SP - Brazil

^[2] Fed Univ ABC, Ctr Math Comp & Cognit, Santo Andre, SP - Brazil

^[3] Univ Fed Sao Carlos, Dept Comp, Sorocaba, SP - Brazil

• Document type: Journal article
• Source: GRAPHS AND COMBINATORICS; v. 35, n. 4, p. 921-931, JUL 2019.
• Web of Science Citations: 0

Abstract

Let k be a positive integer and let D be a digraph. A path partitionP of D is a set of vertex-disjoint paths which coversV(D). Its k-norm is defined as Sigma PPmin[|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 C is orthogonal to a path partition P if each path PP meets min[|V(P)|,k] distinct sets of 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 P is orthogonal to a coloring C if each set CC meets min[|C|,k] paths of 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 vV(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. (AU)

• FAPESP's process: 16/14132-4 - One and two-dimensional bin packing with conflicts and unloading restrictions
• Grantee: Carla Negri Lintzmayer
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• FAPESP's process: 15/11937-9 - Investigation of hard problems from the algorithmic and structural stand points
• Grantee: Flávio Keidi Miyazawa
• Support Opportunities: Research Projects - Thematic Grants