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2015 | 35 | 3 | 557-569
Tytuł artykułu

The Saturation Number for the Length of Degree Monotone Paths

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Wydawca
Rocznik
Tom
35
Numer
3
Strony
557-569
Opis fizyczny
Daty
wydano
2015-08-01
otrzymano
2014-09-14
poprawiono
2014-11-06
zaakceptowano
2014-11-14
online
2015-07-29
Twórcy
autor
autor
Bibliografia
  • [1] B. Bollobás, Extremal Graph Theory (Dover Publications, New York, 2004).
  • [2] Y. Caro, J. Lauri and C. Zarb, Degree monotone paths, ArXiv e-prints (2014) submitted.
  • [3] Y. Caro, J. Lauri and C. Zarb, Degree monotone paths and graph operations, ArXiv e-prints (2014) submitted.
  • [4] J. Deering, Uphill and downhill domination in graphs and related graph parameters, Ph.D. Thesis, ETSU (2013).
  • [5] J. Deering, T.W. Haynes, S.T. Hedetniemi and W. Jamieson, Downhill and uphill domination in graphs, (2013) submitted.
  • [6] J. Deering, T.W. Haynes, S.T. Hedetniemi and W. Jamieson, A Polynomial time algorithm for downhill and uphill domination, (2013) submitted.
  • [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]
  • [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] P. Erdős and G. Szekeres, A combinatorial problem in geometry, Compos. Math. 2 (1935) 463-470.
  • [10] J.R. Faudree, R.J. Faudree and J.R. Schmitt, A survey of minimum saturated graphs, Electron. J. Combin. 18 (2011) #DS19.
  • [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]
  • [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]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1817
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