The Saturation Number for the Length of Degree Monotone Paths

• Autorzy: Yair Caro , Josef Lauri , Christina Zarb
• 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

paths; degrees; saturation.

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2015
• Tom: 35
• Numer: 3
• Strony: 557-569

Daty

wydano

2015-08-01

otrzymano

2014-09-14

poprawiono

2014-11-06

zaakceptowano

2014-11-14

online

2015-07-29

Twórcy

autor

Yair Caro

• Department of Mathematics University of Haifa-Oranim, Israel

autor

Josef Lauri

• Department of Mathematics University of Malta, Malta

autor

Christina Zarb

• Department of Mathematics University of Malta, Malta

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]

Identyfikatory

DOI

10.7151/dmgt.1817