Rainbow cliques in edge-colored graphs. (May 2016)

• Record Type: Journal Article
• Authors: Xu, Chuandong; Hu, Xiaoxue; Wang, Weifan; Zhang, Shenggui
• Is Part Of: European journal of combinatorics. Volume 54(2016:May)
• Page Start: 193
• Page End: 200
• Publication Date: 2016-05
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2015.12.013
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: An edge-colored graph H is called rainbow if e ( H ) = c ( H ), where e ( H ) and c ( H ) are the number of edges of H and colors used in H, respectively. For two graphs G and H, the rainbow number r b ( G, H ) is the minimum number of colors k such that for every edge-coloring of G using k colors, G contains a rainbow H . In this paper we prove that for an edge-colored graph G on n vertices with n ≥ k ≥ 4, if e ( G ) + c ( G ) ≥ ( n 2 ) + t n, k − 2 + 2, then G contains a rainbow clique K k, where t n, k − 2 is the Turán number. This implies the known result r b ( K n, K k ) = t n, k − 2 + 2, and moreover, r b ( G, K k ) ≤ e ( G ¯ ) + r b ( K n, K k ) for n ≥ k ≥ 4 .