Open AccessLinial’s Conjecture for Arc-spine Digraphs
Author(s) -
Lucas Rigo Yoshimura,
Maycon Sambinelli,
Cândida Nunes da Silva,
Orlando Lee
Publication year - 2019
Publication title -
revista eletrônica de iniciação científica
Language(s) - English
Resource type - Journals
ISSN - 1519-8219
DOI - 10.5753/reic.2019.1090
Subject(s) - combinatorics , digraph , disjoint sets , mathematics , partition (number theory) , conjecture , vertex (graph theory) , integer (computer science) , discrete mathematics , graph , computer science , programming language
A path partition P of a digraph D is a collection of directed paths such that every vertex belongs to precisely one path. Given a positive integer k, the k-norm of a path partition P of D is defined as Sum (p in P) min{|p_i|, k}. A path partition of a minimum k-norm is called k-optimal and its k-norm is denoted by π_k(D). A stable set of a digraph D is a subset of pairwise non-adjacentvertices of V(D). Given a positive integer k, we denote by alpha_k(D) the largest set of vertices of D that can be decomposed into k disjoint stable sets of D. In 1981, Linial conjectured that pi_k(D) ≤ alpha_k(D) for every digraph. We say that a digraph D is arc-spine if V(D) can be partitioned into two sets X and Y where X is traceable and Y contains at most one arc in A(D). In this paper we show the validity of Linial’s Conjecture for arc-spine digraphs.