Solutions to conjectures on the (k,ℓ)-rainbow index of complete graphs

• Published in: Networks Vol. 62; no. 3; pp. 220 - 224
• Main Authors: Cai, Qingqiong, Li, Xueliang, Song, Jiangli
• Format: Journal Article
• Language: English
• Published: Hoboken, NJ Blackwell Publishing Ltd 01.10.2013; Wiley; Wiley Subscription Services, Inc
• Subjects: Applied sciences; Computer science; control theory; systems; Exact sciences and technology; Graphs; Information retrieval. Graph; Integers; Networks; Numbers; rainbow connectivity; rainbow index; rainbow tree; Studies; Theoretical computing; Tree(graph); rainbow tree; Graph connectivity; Complete graph; Graph colouring; rainbow connectivity; rainbow index
• ISSN: 0028-3045; 1097-0037
• DOI: 10.1002/net.21513

The (k,ℓ)‐rainbow index rxk,ℓ(G) of a graph G was introduced by Chartrand et al. (Network 54(2) (2009), 75–81; 55 (2010), 360–367). For the complete graph Kn of order n≥6, they showed that rx3,ℓ(Kn)=3 for ℓ=1,2. Furthermore, they conjectured that for every positive integer ℓ, there exists a positive integer N such that rx3,ℓ(Kn)=3 for every integer n≥N. More generally, they conjectured that for every pair of positive integers k and ℓ with k≥3, there exists a positive integer N such that rxk,ℓ(Kn)=k for every integer n≥N. This article provides solutions to these conjectures. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 62(3), 220–224 2013