On greene's theorem for digraphs

Hartman, Irith Ben-Arroyo; Saleh, Fathi; Hershkowitz, Daniel

doi:10.1002/jgt.3190180208

Cited by 13 publications

(13 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Aharoni-Hartman-Hoffman's Conjecture is also known to be valid for bipartite digraphs; the proof is presented in [Hartman et al 1994]. Such result served as one more evidence for the potential validity of Berge's Dual Conjecture for bipartite digraphs.…”

Section: Dual Relationmentioning

confidence: 80%

A proof for Berge’s Dual Conjecture for Bipartite Digraphs

Silva

Lee

2020

REIC

View full text Add to dashboard Cite

Given a (vertex)-coloring $\mathcal{C} = \{C_{1}, C_{2}, ... C_{m}\}$ of a digraph $D$ and a positive integer $k$, the $k$-norm of $\mathcal{C}$ is defined as $ |\mathcal{C}|_k = \sum_{i = 1}^{m} min\{|C_i|, k\}.$ A coloring $\mathcal{C}$ is $k$-optimal if its $k$-norm $|\mathcal{C}|_k$ is minimum over all colorings. A (path) $k$-pack $\mathcal{P}^k$ is a collection of at most $k$ vertex-disjoint paths. A coloring $\mathcal{C}$ and a $k$-pack $\mathcal{P}^k$ are orthogonal if each color class intersects as many paths as possible in $\mathcal{P}^k$, that is, if $|C_i| \ge k$, $|C_i \cap P_j| = 1$ for every path $P_j \in \mathcal{P}^k$, otherwise each vertex of $C_i$ lies in a different path of $\mathcal{P}^k$. In 1982, Berge conjectured that for every $k$-optimal coloring $\mathcal{C}$ there is a $k$-pack $\mathcal{P}^k$ orthogonal to $\mathcal{C}$. This conjecture is false for arbitrary digraphs, having a counterexample with odd cycle. In this paper we prove this conjecture for bipartite digraphs. In addition we show that the conjecture cannot hold for perfect graphs by exhibiting a counterexample.

show abstract

Section: Dual Relationmentioning

confidence: 80%

A proof for Berge’s Dual Conjecture for Bipartite Digraphs

Silva

Lee

2020

REIC

View full text Add to dashboard Cite

show abstract

“…We denote by D Here we prove Linial's Dual Conjecture for path-spine digraphs. One important partial result for this conjecture is due to Hartman, Saleh and Hershkowits, who proved in [Hartman et al 1994] that it holds for split digraphs (such proof can be easily adapted for spine digraphs). Therefore, our result is, as far as we know, the first generalization of Hartman, Saleh and Hershkowits's proof for split digraphs since its publication in 1994.…”

Section: Introductionmentioning

confidence: 99%

Linial's Dual Conjecture for Path-Spine Digraphs

Carvalho¹,

Silva²,

Lee³

2020

Anais Do Encontro De Teoria Da Computação (ETC 2020)

View full text Add to dashboard Cite

Given a digraph D, a coloring 𝒞 of D is a partition of V(D) into stable sets. The k-norm of 𝒞 is defined as ΣC∈𝒞 min{|C|, k}. A coloring of D with minimum k-norm has its k-norm noted by χk(D). A (path)-k-pack of a digraph D is a set of k vertex-disjoint (directed) paths of D. The weight of a k-pack is the number of vertices covered by the k-pack. We denote by λk(D) the weight of a maximum k-pack. Linial conjectured that χk(D) ≤ λk(D) for every digraph. Such conjecture remains open, but has been proved for some classes of digraphs. We prove the conjecture for path-spine digraphs, defined as follows. A digraph D is path-spine if there exists a partition {X, Y} of V(D) such that D[X] has a Hamilton path and every arc in D[Y] belongs to a single path Q.

show abstract

“…A digraph D is a spine digraph if there exists a partition {X, Y } of V (D) such that D[X] is traceable and Y is a stable set in D. Spine digraphs are a superclass of split digraphs. Long before, in 1994, Hartman, Saleh and Hershkowitz [Hartman et al 1994] gave a proof of a different (although related) conjecture of Linial which we refer to as Linial's Dual Conjecture (see [Sambinelli 2018] for its statement). The proof of Sambinelli, Nunes da Silva and Lee [Sambinelli et al 2017] has some similarity in structure to that of Hartman, Saleh and Hershkowitz; however some particular technique had to be developed to address Linial's Conjecture.…”

Section: Introductionmentioning

confidence: 99%

Linial’s Conjecture for Arc-spine Digraphs

Yoshimura

Sambinelli

Silva

et al. 2019

REIC

View full text Add to dashboard Cite

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.

show abstract

On greene's theorem for digraphs

Cited by 13 publications

References 9 publications

A proof for Berge’s Dual Conjecture for Bipartite Digraphs

A proof for Berge’s Dual Conjecture for Bipartite Digraphs

Linial's Dual Conjecture for Path-Spine Digraphs

Linial’s Conjecture for Arc-spine Digraphs

Contact Info

Product

Resources

About