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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1068,
author = {Izak Broere and Michael Dorfling and Jean E. Dunbar and Marietjie Frick},
title = {A path(ological) partition problem},
journal = {Discussiones Mathematicae Graph Theory},
volume = {18},
year = {1998},
pages = {113-125},
zbl = {0912.05048},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1068}
}
Izak Broere; Michael Dorfling; Jean E. Dunbar; Marietjie Frick. A path(ological) partition problem. Discussiones Mathematicae Graph Theory, Tome 18 (1998) pp. 113-125. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1068/
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