Rainbow Connection In Sparse Graphs

Abstract An edge-coloured connected graph G = (V,E) is called rainbow-connected if each pair of distinct vertices of G is connected by a path whose edges have distinct colours. The rainbow connection number of G, denoted by rc(G), is the minimum number of colours such that G is rainbow-connected. In this paper we prove that rc(G) ≤ k if |V (G)| = n and for all integers n and k with n − 6 ≤ k ≤ n − 3. We also show that this bound is tight.

[1]  L. Sunil Chandran,et al.  Rainbow connection number and connected dominating sets , 2010, J. Graph Theory.

[2]  Xueliang Li,et al.  Rainbow Connection Number of Dense Graphs , 2013, Discuss. Math. Graph Theory.

[3]  Zsófia Osváth,et al.  DOI: 10 , 2011 .

[4]  Raphael Yuster,et al.  The rainbow connection of a graph is (at most) reciprocal to its minimum degree , 2010, J. Graph Theory.

[5]  Helena Reimer-Burgrova.  A matter of security. , 2021, Biomedical instrumentation & technology.

[6]  Raphael Yuster,et al.  Hardness and Algorithms for Rainbow Connectivity , 2009, STACS.

[7]  L. Lovász.  A homology theory for spanning tress of a graph , 1977 .

[8]  Garry L. Johns,et al.  Rainbow connection in graphs , 2008 .

[9]  Ingo Schiermeyer,et al.  Graphs with rainbow connection number two , 2011, Discuss. Math. Graph Theory.

[10]  Ingo Schiermeyer.  Rainbow Connection in Graphs with Minimum Degree Three , 2009, IWOCA.

[11]  Xueliang Li,et al.  Rainbow Connections of Graphs , 2012 .

[12]  Raphael Yuster,et al.  Hardness and algorithms for rainbow connection , 2008, J. Comb. Optim..

[13]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[14]  Raphael Yuster,et al.  On Rainbow Connection , 2008, Electron. J. Comb..