TY - JOUR

T1 - The rainbow connection of a graph is (at most) reciprocal to its minimum degree

AU - Krivelevich, Michael

AU - Yuster, Raphael

PY - 2010/3

Y1 - 2010/3

N2 - An edge-colored graph G is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow edge-connected. We prove that if G has n vertices and minimum degreeδ then rc(G)<20n /δ. This solves open problems from Y. Caro, A. Lev Y. Roditty, Z. Tuza, and R. Yuster (Electron J Combin 15 (2008), #R57) and S. Chakrborty, E. Fischer, A. Matsliah, and R. Yuster (Hardness and algorithms for rainbow connectivity, Freiburg (2009), pp. 243-254). A vertex-colored graph G is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. One cannot upper-bound one of these parameters in terms of the other. Nevertheless, we prove that if G has n vertices and minimum degree δthen rvc(G)<11n /δ We note that the proof in this case is different from the proof for the edge-colored case, and we cannot deduce one from the other.

AB - An edge-colored graph G is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow edge-connected. We prove that if G has n vertices and minimum degreeδ then rc(G)<20n /δ. This solves open problems from Y. Caro, A. Lev Y. Roditty, Z. Tuza, and R. Yuster (Electron J Combin 15 (2008), #R57) and S. Chakrborty, E. Fischer, A. Matsliah, and R. Yuster (Hardness and algorithms for rainbow connectivity, Freiburg (2009), pp. 243-254). A vertex-colored graph G is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. One cannot upper-bound one of these parameters in terms of the other. Nevertheless, we prove that if G has n vertices and minimum degree δthen rvc(G)<11n /δ We note that the proof in this case is different from the proof for the edge-colored case, and we cannot deduce one from the other.

KW - Minimum degree

KW - Rainbow connection

UR - http://www.scopus.com/inward/record.url?scp=76649090640&partnerID=8YFLogxK

U2 - 10.1002/jgt.20418

DO - 10.1002/jgt.20418

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AN - SCOPUS:76649090640

SN - 0364-9024

VL - 63

SP - 185

EP - 191

JO - Journal of Graph Theory

JF - Journal of Graph Theory

IS - 3

ER -