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2013 | 33 | 2 | 411-428
Tytuł artykułu

On Closed Modular Colorings of Trees

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Two vertices u and v in a nontrivial connected graph G are twins if u and v have the same neighbors in V (G) − {u, v}. If u and v are adjacent, they are referred to as true twins; while if u and v are nonadjacent, they are false twins. For a positive integer k, let c : V (G) → Zk be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex coloring c′ : V (G) → Zk defined by c′(v) = P u∈N[v] c(u) for each v ∈ V (G), where N[v] is the closed neighborhood of v. Then c is called a closed modular k-coloring if c′(u) 6= c′(v) in Zk for all pairs u, v of adjacent vertices that are not true twins. The minimum k for which G has a closed modular k-coloring is the closed modular chromatic number mc(G) of G. The closed modular chromatic number is investigated for trees and determined for several classes of trees. For each tree T in these classes, it is shown that mc(T) = 2 or mc(T) = 3. A closed modular k-coloring c of a tree T is called nowhere-zero if c(x) 6= 0 for each vertex x of T. It is shown that every tree of order 3 or more has a nowhere-zero closed modular 4-coloring.
Wydawca
Rocznik
Tom
33
Numer
2
Strony
411-428
Opis fizyczny
Daty
wydano
2013-05-01
online
2013-04-13
Twórcy
  • Department of Mathematics Western Michigan University Kalamazoo, MI 49008, USA
autor
  • Department of Mathematics Western Michigan University Kalamazoo, MI 49008, USA, ping.zhang@wmich.edu
Bibliografia
  • [1] L. Addario-Berry, R.E.L. Aldred, K. Dalal and B.A. Reed, Vertex colouring edge partitions, J. Combin. Theory (B) 94 (2005) 237-244. doi:10.1016/j.jctb.2005.01.001[Crossref]
  • [2] G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz and F. Saba, Irregular networks, Congr. Numer. 64 (1988) 197-210.
  • [3] G. Chartrand, F. Okamoto and P. Zhang, The sigma chromatic number of a graph, Graphs Combin. 26 (2010) 755-773. doi:10.1007/s00373-010-0952-7[Crossref]
  • [4] G. Chartrand, B. Phinezy and P. Zhang, On closed modular colorings of regular graphs, Bull. Inst. Combin. Appl. 66 (2012) 7-32.
  • [5] G. Chartrand, L. Lesniak and P. Zhang, Graphs & Digraphs, 5th Edition (Chapman & Hall/CRC, Boca Raton, 2010).
  • [6] G. Chartrand and P. Zhang, Chromatic Graph Theory (Chapman & Hall/CRC, Boca Raton, 2008). doi:10.1201/9781584888017[Crossref]
  • [7] H. Escuadro, F. Okamoto and P. Zhang, Vertex-distinguishing colorings of graphs- A survey of recent developments, AKCE Int. J. Graphs Comb. 4 (2007) 277-299.
  • [8] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 16 (2009) #.DS6
  • [9] R.B. Gnana Jothi, Topics in Graph Theory, Ph.D. Thesis, Madurai Kamaraj University 1991.
  • [10] S.W. Golomb, How to number a graph, in: Graph Theory and Computing (Academic Press, New York, 1972) 23-37.
  • [11] R. Jones, K. Kolasinski and P. Zhang, A proof of the modular edge-graceful trees conjecture, J. Combin. Math. Combin. Comput., to appear.
  • [12] M. Karónski, T. Luczak and A. Thomason, Edge weights and vertex colours, J. Combin. Theory (B) 91 (2004) 151-157. doi:10.1016/j.jctb.2003.12.001[Crossref]
  • [13] A. Rosa, On certain valuations of the vertices of a graph, in: Theory of Graphs, Pro. Internat. Sympos. Rome 1966 (Gordon and Breach, New York, 1967) 349-355.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1678
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