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2014 | 34 | 2 | 233-247
Tytuł artykułu

The depression of a graph and k-kernels

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
An edge ordering of a graph G is an injection f : E(G) → R, the set of real numbers. A path in G for which the edge ordering f increases along its edge sequence is called an f-ascent ; an f-ascent is maximal if it is not contained in a longer f-ascent. The depression of G is the smallest integer k such that any edge ordering f has a maximal f-ascent of length at most k. A k-kernel of a graph G is a set of vertices U ⊆ V (G) such that for any edge ordering f of G there exists a maximal f-ascent of length at most k which neither starts nor ends in U. Identifying a k-kernel of a graph G enables one to construct an infinite family of graphs from G which have depression at most k. We discuss various results related to the concept of k-kernels, including an improved upper bound for the depression of trees.
Wydawca
Rocznik
Tom
34
Numer
2
Strony
233-247
Opis fizyczny
Daty
wydano
2014-05-01
online
2014-04-12
Twórcy
autor
  • Department of Mathematics and Statistics University of Victoria, P.O. Box 3060 STN CSC Victoria, BC, Canada V8W 3R4, mschurch@uvic.ca
  • Department of Mathematics and Statistics University of Victoria, P.O. Box 3060 STN CSC Victoria, BC, Canada V8W 3R4, mynhardt@math.uvic.ca
Bibliografia
  • [1] A. Bialostocki and Y. Roditty, A monotone path in an edge-ordered graph, Int. J. Math. Math. Sci. 10 (1987) 315-320. doi:10.1155/S0161171287000383[Crossref]
  • [2] A.P. Burger, E.J. Cockayne and C.M. Mynhardt, Altitude of small complete and complete bipartite graphs, Australas. J. Combin. 31 (2005) 167-177.
  • [3] A.R. Calderbank and F.R.K. Chung and D.G. Sturtevant, Increasing sequences with nonzero block sums and increasing paths in edge-ordered graphs, Discrete Math. 50 (1984) 15-28. doi:10.1016/0012-365X(84)90031-1[Crossref]
  • [4] G. Chartrand and L.M. Lesniak, Graphs and Digraphs (Third Ed.) (Wadsworth, 1996).
  • [5] V. Chv´atal and J. Koml´os, Some combinatorial theorems on monotonicity, Canad. Math. Bull. 14 (1971) 151-157. doi:10.4153/CMB-1971-028-8[Crossref]
  • [6] T.C. Clark and B. Falvai, N.D.R. Henderson and C.M. Mynhardt, Altitude of 4- regular circulants, AKCE Int. J. Graphs Comb. 1 (2004) 149-166.
  • [7] E.J. Cockayne and G. Geldenhuys, P.J.P Grobler, C.M. Mynhardt and J.H. van Vuuren, The depression of a graph, Util. Math. 69 (2006) 143-160.
  • [8] E.J. Cockayne and C.M. Mynhardt, Altitude of K3,n, J. Combin. Math. Combin. Comput. 52 (2005) 143-157.
  • [9] E.J. Cockayne and C.M. Mynhardt, A lower bound for the depression of trees, Aus- tralas. J. Combin. 35 (2006) 319-328.
  • [10] T. Dzido and H. Furma´nczyk, Altitude of wheels and wheel-like graphs, Cent. Eur. J. Math. 8 (2010) 319-326. doi:10.7151/s11533-010-0017-4[WoS][Crossref]
  • [11] I. Gaber-Rosenblum and Y. Roditty, The depression of a graph and the diameter of its line graph, Discrete Math. 309 (2009) 1774-1778. doi:10.1016/j.disc.2008.03.002[Crossref][WoS]
  • [12] R.L. Graham and D.J. Kleitman, Increasing paths in edge ordered graphs, Period. Math. Hungar. 3 (1973) 141-148. doi:10.1007/BF02018469[Crossref]
  • [13] J. Katreniˇc and G. Semaniˇsin, Finding monotone paths in edge-ordered graphs, Dis- crete Appl. Math. 158 (2010) 1624-1632. doi:10.1016/j.dam.2010.05.018[Crossref]
  • [14] C.M. Mynhardt, Trees with depression three, Discrete Math. 308 (2008) 855-864. doi:10.1016/j.disc.2007.07.039[Crossref][WoS]
  • [15] C.M. Mynhardt, A.P. Burger and T.C. Clark, B. Falvai, and N.D.R. Henderson, Altitude of regular graphs with girth at least five, Discrete Math. 294 (2005) 241-257. doi:10.1016/j.disc.2005.02.007[Crossref]
  • [16] C.M. Mynhardt and M. Schurch, A class of graphs with depression three, Discrete Math. 313 (2013) 1224-1232. doi:10.1016/j.disc.2012.05.011[WoS][Crossref]
  • [17] Y. Roditty, B. Shoham and R. Yuster, Monotone paths in edge-ordered sparse graphs, Discrete Math. 226 (2001) 411-417. doi:10.1016/S0012-365X(00)00174-6[Crossref]
  • [18] R. Yuster, Large monotone paths in graphs with bounded degree, Graphs Combin. 17 (2001) 579-587. doi:10.1007/s003730170031 [Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1736
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