# A Survey of the Path Partition Conjecture

@inproceedings{Frick2013ASO,
title={A Survey of the Path Partition Conjecture},
author={Marietjie Frick},
booktitle={Discuss. Math. Graph Theory},
year={2013}
}
• M. Frick
• Published in Discuss. Math. Graph Theory 1 March 2013
• Mathematics, Computer Science
Abstract 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 PPC that have appeared in the literature since its first formulation in 1981.
3 Citations
Extended Path Partition Conjecture for Semicomplete and Acyclic Compositions
• Computer Science, Mathematics
ArXiv
• 2021
A conjecture stronger than PPC is introduced using a property first studied by Bang-Jensen, Nielsen and Yeo (2006) and it is shown that the stronger conjecture holds for wide families of acyclic and semicomplete compositions.
My Top 10 Graph Theory Conjectures and Open Problems
This paper presents brief discussions of ten of my favorite, well-known, and not so well-known conjectures and open problems in graph theory, including (1) the 1963 Vizing’s Conjecture about the
The Path Partition Conjecture is True and its Validity Yields Upper Bounds for Detour Chromatic Number and Star Chromatic Number
The detour order of a graph $G$, denoted $\tau(G)$, is the order of a longest path in $G$. A partition $(A, B)$ of $V(G)$ such that $\tau(\langle A \rangle) \leq a$ and \$\tau(\langle B \rangle) \leq

## References

SHOWING 1-10 OF 79 REFERENCES
The directed path partition conjecture
• Mathematics, Computer Science
Discuss. Math. Graph Theory
• 2005
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
The Path Partition Conjecture is true for claw-free graphs
• Computer Science, Mathematics
Discret. Math.
• 2007
On a cycle partition problem
• M. Nielsen
• Computer Science, Mathematics
Discret. Math.
• 2008
Path Partitions and Pn-free Sets
• Mathematics, Computer Science
Electron. Notes Discret. Math.
• 2002
Abstract The detour order τ(G) of a graph G is the order of a longest path of G. A partition (A, B) of V is called an (a, b)-partition of G if τ(G[A]) ≤ a and τ(G[B]) ≤ b. The Path Partition
Graphs with not all possible path-kernels
• Computer Science, Mathematics
Discret. Math.
• 2004
Longest path partitions in generalizations of tournaments
• Mathematics, Computer Science
Discret. Math.
• 2006
A path(ological) partition problem
• Mathematics, Computer Science
Discuss. Math. Graph Theory
• 1998
It is shown that several classes of graphs have this partition property and 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.
A note on a cycle partition problem
• Mathematics, Computer Science
Appl. Math. Lett.
• 2011
A note on the Path Kernel Conjecture
• Computer Science, Mathematics
Discret. Math.
• 2009