A degree monotone path in a graph G is a path P such that the sequence of degrees of the vertices in the order in which they appear on P is monotonic. The length (number of vertices) of the longest degree monotone path in G is denoted by mp(G). This parameter, inspired by the well-known Erdős- Szekeres theorem, has been studied by the authors in two earlier papers. Here we consider a saturation problem for the parameter mp(G). We call G saturated if, for every edge e added to G, mp(G + e) > mp(G), and we define h(n, k) to be the least possible number of edges in a saturated graph G on n vertices with mp(G) < k, while mp(G+e) ≥ k for every new edge e. We obtain linear lower and upper bounds for h(n, k), we determine exactly the values of h(n, k) for k = 3 and 4, and we present constructions of saturated graphs.
@article{bwmeta1.element.doi-10_7151_dmgt_1817,
author = {Yair Caro and Josef Lauri and Christina Zarb},
title = {The Saturation Number for the Length of Degree Monotone Paths},
journal = {Discussiones Mathematicae Graph Theory},
volume = {35},
year = {2015},
pages = {557-569},
zbl = {1317.05030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1817}
}
Yair Caro; Josef Lauri; Christina Zarb. The Saturation Number for the Length of Degree Monotone Paths. Discussiones Mathematicae Graph Theory, Tome 35 (2015) pp. 557-569. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1817/
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