Berge's conjecture on directed path partitions—a survey
نویسندگان
چکیده
منابع مشابه
Berge's conjecture on directed path partitions - a survey
Berge’s conjecture from 1982 on path partitions in directed graphs generalizes and extends Dilworth’s Theorem and the Greene-Kleitman Theorem which are well known for partially ordered sets. The conjecture relates path partitions to a collection of k independent sets, for each k ≥ 1. The conjecture is still open and intriguing for all k > 1. In this paper, we will survey partial results on the ...
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The Directed Path Partition Conjecture is the following: If D is a digraph that contains no path with more than λ vertices then, for every pair (a, b) of positive integers with λ = a + b, there exists a vertex partition (A, B) of D such that no path in D〈A〉 has more than a vertices and no path in D〈B〉 has more than b vertices.We develop methods for finding the desired partitions for various cla...
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The Path Partition Conjecture (PPC) states that if G is any graph and (λ1, λ2) any pair of positive integers such that G has no path with more than λ1 + λ2 vertices, then there exists a partition (V1, V2) of the vertex set of G such that Vi has no path with more than λi vertices, i = 1, 2. We present a brief history of the PPC, discuss its relation to other conjectures and survey results on the...
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An oriented perfect path double cover (OPPDC) of a graph $G$ is a collection of directed paths in the symmetric orientation $G_s$ of $G$ such that each arc of $G_s$ lies in exactly one of the paths and each vertex of $G$ appears just once as a beginning and just once as an end of a path. Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete Math. 276 (2004) 287-294) conjectured that ...
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an oriented perfect path double cover (oppdc) of a graph $g$ is a collection of directed paths in the symmetric orientation $g_s$ of $g$ such that each arc of $g_s$ lies in exactly one of the paths and each vertex of $g$ appears just once as a beginning and just once as an end of a path. maxov{'a} and ne{v{s}}et{v{r}}il (discrete math. 276 (2004) 287-294) conjectured that ...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2006
ISSN: 0012-365X
DOI: 10.1016/j.disc.2005.12.039