The Saturation Number for the Length of Degree Monotone Paths

Yair Caro; Josef Lauri; Christina Zarb

Discussiones Mathematicae Graph Theory (2015)

  • Volume: 35, Issue: 3, page 557-569
  • ISSN: 2083-5892

Abstract

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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.

How to cite

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Yair Caro, Josef Lauri, and Christina Zarb. "The Saturation Number for the Length of Degree Monotone Paths." Discussiones Mathematicae Graph Theory 35.3 (2015): 557-569. <http://eudml.org/doc/271208>.

@article{YairCaro2015,
abstract = {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.},
author = {Yair Caro, Josef Lauri, Christina Zarb},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {paths; degrees; saturation.; saturation},
language = {eng},
number = {3},
pages = {557-569},
title = {The Saturation Number for the Length of Degree Monotone Paths},
url = {http://eudml.org/doc/271208},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Yair Caro
AU - Josef Lauri
AU - Christina Zarb
TI - The Saturation Number for the Length of Degree Monotone Paths
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 3
SP - 557
EP - 569
AB - 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.
LA - eng
KW - paths; degrees; saturation.; saturation
UR - http://eudml.org/doc/271208
ER -

References

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  1. [1] B. Bollobás, Extremal Graph Theory (Dover Publications, New York, 2004). Zbl1099.05044
  2. [2] Y. Caro, J. Lauri and C. Zarb, Degree monotone paths, ArXiv e-prints (2014) submitted. Zbl1317.05030
  3. [3] Y. Caro, J. Lauri and C. Zarb, Degree monotone paths and graph operations, ArXiv e-prints (2014) submitted. 
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  7. [7] M. Eliáš and J. Matoušek, Higher-order Erdős-Szekeres theorems, Adv. Math. 244 (2013) 1-15. doi:10.1016/j.aim.2013.04.020[Crossref] Zbl1283.05175
  8. [8] P. Erdős, A. Hajnal and J.W. Moon, A problem in graph theory, Amer. Math.Monthly 71 (1964) 1107-1110. doi:10.2307/2311408[Crossref] 
  9. [9] P. Erdős and G. Szekeres, A combinatorial problem in geometry, Compos. Math. 2 (1935) 463-470. Zbl0012.27010
  10. [10] J.R. Faudree, R.J. Faudree and J.R. Schmitt, A survey of minimum saturated graphs, Electron. J. Combin. 18 (2011) #DS19. Zbl1222.05113
  11. [11] T.W. Haynes, S.T. Hedetniemi, J.D. Jamieson and W.B. Jamieson, Downhill dom- ination in graphs, Discuss. Math. Graph Theory 34 (2014) 603-612. doi:10.7151/dmgt.1760[Crossref] Zbl1295.05176
  12. [12] L. Kászonyi and Zs. Tuza, Saturated graphs with minimal number of edges, J. Graph Theory 10 (1986) 203-210. doi:10.1002/jgt.3190100209 [Crossref] Zbl0593.05041

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