The 3-Rainbow Index of a Graph

• Autorzy: Lily Chen , Xueliang Li , Kang Yang , Yan Zhao
• Języki publikacji: EN

Abstrakty

EN

Let G be a nontrivial connected graph with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ ℕ, where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. For a vertex subset S ⊆ V (G), a tree that connects S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a rainbow S-tree for each k-subset S of V (G) is called the k-rainbow index of G, denoted by rxk(G). In this paper, we first determine the graphs of size m whose 3-rainbow index equals m, m − 1, m − 2 or 2. We also obtain the exact values of rx3(G) when G is a regular multipartite complete graph or a wheel. Finally, we give a sharp upper bound for rx3(G) when G is 2-connected and 2-edge connected. Graphs G for which rx3(G) attains this upper bound are determined.

Słowa kluczowe

EN

rainbow tree; S-tree; k-rainbow index

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2015
• Tom: 35
• Numer: 1
• Strony: 81-94

Daty

wydano

2015-02-01

otrzymano

2013-12-27

poprawiono

2013-12-27

zaakceptowano

2014-02-12

online

2015-02-06

Twórcy

autor

Lily Chen

• Center for Combinatorics and LPMC-TJKLC Nankai University Tianjin 300071, China

autor

Xueliang Li

• Center for Combinatorics and LPMC-TJKLC Nankai University Tianjin 300071, China

autor

Kang Yang

• Center for Combinatorics and LPMC-TJKLC Nankai University Tianjin 300071, China

autor

Yan Zhao

• Center for Combinatorics and LPMC-TJKLC Nankai University Tianjin 300071, China

Bibliografia

• [1] J.A. Bondy and U.S.R. Murty, Graph Theory (GTM 244, Springer, 2008).
• [2] Y. Caro, A. Lev, Y. Roditty, Zs. Tuza and R. Yuster, On rainbow connection, Electron. J. Combin. 15(1) (2008) R57.
• [3] G. Chartrand, G. Johns, K. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohem. 133 (2008) 85-98.
• [4] G. Chartrand, F. Okamoto and P. Zhang, Rainbow trees in graphs and generalized connectivity, Networks 55 (2010) 360-367. doi:10.1002/net.20339[WoS][Crossref]
• [5] G. Chartrand, G. Johns, K. McKeon and P. Zhang, The rainbow connectivity of a graph, Networks 54(2) (2009) 75-81. doi:10.1002/net.20296[Crossref][WoS]
• [6] X. Li and Y. Sun, Rainbow Connections of Graphs (Springer Briefs in Math., Springer, 2012).
• [7] X. Li, Y. Shi and Y. Sun, Rainbow connections of graphs: A survey, Graphs Combin. 29 (2013) 1-38. doi:10.1007/s00373-012-1243-2[WoS][Crossref]

Identyfikatory

DOI

10.7151/dmgt.1780