Geodesic
The Saturation Number for the Length of Degree Monotone Paths
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 3, pp. 557-569

Voir la notice de l'article provenant de la source Library of Science

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) gt; 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) lt; 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.
Keywords: paths, degrees, saturation 
@article{DMGT_2015_35_3_a12,
     author = {Caro, Yair and Lauri, Josef and Zarb, Christina},
     title = {The {Saturation} {Number} for the {Length} of {Degree} {Monotone} {Paths}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {557--569},
     publisher = {mathdoc},
     volume = {35},
     number = {3},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2015_35_3_a12/}
}
TY  - JOUR
AU  - Caro, Yair
AU  - Lauri, Josef
AU  - Zarb, Christina
TI  - The Saturation Number for the Length of Degree Monotone Paths
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2015
SP  - 557
EP  - 569
VL  - 35
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2015_35_3_a12/
LA  - en
ID  - DMGT_2015_35_3_a12
ER  - 
%0 Journal Article
%A Caro, Yair
%A Lauri, Josef
%A Zarb, Christina
%T The Saturation Number for the Length of Degree Monotone Paths
%J Discussiones Mathematicae. Graph Theory
%D 2015
%P 557-569
%V 35
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2015_35_3_a12/
%G en
%F DMGT_2015_35_3_a12
Caro, Yair; Lauri, Josef; Zarb, Christina. The Saturation Number for the Length of Degree Monotone Paths. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 3, pp. 557-569. http://geodesic.mathdoc.fr/item/DMGT_2015_35_3_a12/