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2012 | 32 | 1 | 63-80
Tytuł artykułu

Vertex rainbow colorings of graphs

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In a properly vertex-colored graph G, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P. If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring of G. The minimum number of colors needed in a vertex rainbow coloring of G is the vertex rainbow connection number vrc(G) of G. Thus if G is a connected graph of order n ≥ 2, then 2 ≤ vrc(G) ≤ n. We present characterizations of all connected graphs G of order n for which vrc(G) ∈ {2,n-1,n} and study the relationship between vrc(G) and the chromatic number χ(G) of G. For a connected graph G of order n and size m, the number m-n+1 is the cycle rank of G. Vertex rainbow connection numbers are determined for all connected graphs of cycle rank 0 or 1 and these numbers are investigated for connected graphs of cycle rank 2.
Wydawca
Rocznik
Tom
32
Numer
1
Strony
63-80
Opis fizyczny
Daty
wydano
2012
otrzymano
2010-06-15
poprawiono
2011-01-20
zaakceptowano
2011-01-24
Twórcy
  • Mathematics Department, University of Wisconsin La Crosse, La Crosse, WI 54601, USA
  • Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA
autor
  • Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA
autor
  • Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA
Bibliografia
  • [1] G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohem. 133 (2008) 85-98.
  • [2] G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, The rainbow connectivity of a graph, Networks 54 (2009) 75-81, doi: 10.1002/net.20296.
  • [3] G. Chartrand, F. Okamoto and P. Zhang, Rainbow trees in graphs and generalized connectivity, Networks 55 (2010) 360-367.
  • [4] G. Chartrand and P. Zhang, Chromatic Graph Theory (Chapman & Hall/CRC Press, 2009).
  • [5] M. Krivelevich and R. Yuster,The rainbow connection of a graph is (at most) reciprocal to its minimum degree, J. Graph Theory 63 (2010) 185-191
  • [6] H. Whitney, Congruent graphs and the connectivity of graphs, Amer. J. Math. 54 (1932) 150-168, doi: 10.2307/2371086.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1586
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