# A proof for Berges Dual Conjecture for Bipartite Digraphs

• Caroline Silva UFSCAR
• Cândida Silva UFSCAR
• Orlando Lee UNICAMP

### Resumo

Given a (vertex)-coloring C = C 1 , C 2 , ...C m of P a digraph D and a positive integer k, the k-norm of C is defined as C k = m i=1 min C i, k. A coloring C is k-optimal if its k-norm C k is minimum over all colorings. A (path) k-pack P k is a collection of at most k vertex-disjoint paths. A coloring C and a k-pack P k are orthogonal if each color class intersects as many paths as possible in P k , that is, if C i k, C i P j = 1 for every path P j P k , otherwise each vertex of C i lies in a different path of P k . In 1982, Berge conjectured that for every k-optimal coloring C there is a k-pack P k orthogonal to C. This conjecture is false for arbitrary digraphs, having a counterexample with odd cycle. In this paper we prove this conjecture for bipartite digraphs.

Palavras-chave: Berges Dual Conjecture, Bipartite Digraphs

### Referências

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Publicado
02/07/2019
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SILVA, Caroline; SILVA, Cândida ; LEE, Orlando . A proof for Berges Dual Conjecture for Bipartite Digraphs. In: ENCONTRO DE TEORIA DA COMPUTAÇÃO (ETC), 4. , 2019, Belém. Anais [...]. Porto Alegre: Sociedade Brasileira de Computação, 2019 . ISSN 2595-6116. DOI: https://doi.org/10.5753/etc.2019.6398.