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On greene's theorem for digraphs
Author(s) -
Hartman Irith BenArroyo,
Saleh Fathi,
Hershkowitz Daniel
Publication year - 1994
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.3190180208
Subject(s) - mathematics , combinatorics , conjecture , bipartite graph , cardinality (data modeling) , discrete mathematics , path (computing) , graph , computer science , data mining , programming language
Greene's Theorem states that the maximum cardinality of an optimal k ‐path in a poset is equal to the minimum k ‐norm of a k ‐optimal coloring. This result was extended to all acyclic digraphs, and is conjectured to hold for general digraphs. We prove the result for general digraphs in which an optimal k ‐path contains a path of cardinality one. This implies the validity of the conjecture for all bipartite digraphs. We also extend Greene's Theorem to all split graphs.