Elsevier

Discrete Mathematics

Volume 313, Issue 3, 6 February 2013, Pages 293-300
Discrete Mathematics

The Path Partition Conjecture is true for some generalizations of tournaments

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Abstract

The Path Partition Conjecture for digraphs states that for every digraph D, and every choice of positive integers λ1,λ2 such that λ1+λ2 equals the order of a longest directed path in D, there exists a partition of D in two subdigraphs D1,D2 such that the order of the longest path in Di is at most λi for i=1,2.

We present sufficient conditions for a digraph to satisfy the Path Partition Conjecture. Using these results, we prove that strong path mergeable, arc-locally semicomplete, strong 3-quasi-transitive, strong arc-locally in-semicomplete and strong arc-locally out-semicomplete digraphs satisfy the Path Partition Conjecture. Some previous results are generalized.

Keywords

Digraph
Longest path
Path Partition Conjecture
Tournament
Nearly tournament

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