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The directed path partition conjecture

Marietjie Frick;  Susan van Aardt;  Gcina Dlamini;  Jean Dunbar;  Ortrud Oellermann — 2005

Discussiones Mathematicae Graph Theory

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 classes of digraphs.

A path(ological) partition problem

Izak Broere;  Michael Dorfling;  Jean E. Dunbar;  Marietjie Frick — 1998

Discussiones Mathematicae Graph Theory

Let τ(G) denote the number of vertices in a longest path of the graph G and let k₁ and k₂ be positive integers such that τ(G) = k₁ + k₂. The question at hand is whether the vertex set V(G) can be partitioned into two subsets V₁ and V₂ such that τ(G[V₁] ) ≤ k₁ and τ(G[V₂] ) ≤ k₂. We show that several classes of graphs have this partition property.

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