Zata Yumnı AWANIS, Anm SALMAN, Suhadı Wıdo SAPUTRO, Martın BACA, Andrea SEMANICOVA-FENOVCIKOVA

The strong 3-rainbow index of edge-amalgamation of some graphs

Let G be a nontrivial, connected, and edge-colored graph of order n ≥ 3, where adjacent edges may be colored the same. Let k be an integer with 2 ≤ k ≤ n. A tree T in G is a rainbow tree if no two edges of T are colored the same. For S ⊆ V G , the Steiner distance d S of S is the minimum size of a tree in G containing S . An edge-coloring of G is called a strong k -rainbow coloring if for every set S of k vertices of G there exists a rainbow tree of size d S in G containing S . The minimum number of colors needed in a strong k -rainbow coloring of G is called the strong k -rainbow index srxk G of G. In this paper, we study the strong 3-rainbow index of edge-amalgamation of graphs. We provide a sharp upper bound for the srx3 of edge-amalgamation of graphs. We also determine the srx3 of edge-amalgamation of some graphs.

Keywords:

Edge-amalgamation, rainbow coloring, rainbow tree, strong k -rainbow index,

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[1] Awanis ZY, Salman A. The 3-rainbow index of amalgamation of some graphs with diameter 2. Journal of Physics: Conference Series 2019; 1127: 012058. doi: 10.1088/1742-6596/1127/1/012058
[2] Chartrand G, Johns GL, McKeon KA, Zhang P. Rainbow connection in graphs. Mathematica Bohemica 2008; 133: 85-98.
[3] Chartrand G, Okamoto F, Zhang P. Rainbow trees in graphs and generalized connectivity. Networks 2010; 55: 360-367. doi: 10.1002/net.20339
[4] Chen L, Li X, Yang K, Zhao Y. The 3-rainbow index of a graph. Discussiones Mathematicae Graph Theory 2015; 35 (1): 81-94.
[5] Diestel R. Graph Theory. 4th Edition. Heidelberg, Germany: Springer, 2010.
[6] Fitriani D, Salman A. Rainbow connection number of amalgamation of some graphs. AKCE International Journal of Graphs and Combinatorics 2016; 13: 90-99. doi: 10.1016/j.akcej.2016.03.004
[7] Kumala IS, Salman A. The rainbow connection number of a flower (Cm, Kn) graph and a flower (C3, Fn) graph. Procedia Computer Science 2015; 74: 168-172. doi: 10.1016/j.procs.2015.12.094
[8] Li X, Shi Y, Sun Y. Rainbow connections of graphs: a survey. Graphs and Combinatorics 2013; 29 (1): 1-38.
[9] Li X, Sun Y. An updated survey on rainbow connections of graphs - a dynamic survey. Theory and Applications of Graphs 2017; 0 (1): 3. doi: 10.20429/tag.2017.000103
[10] Liu T, Hu Y. The 3-rainbow index of graph operations. WSEAS Transactions on Mathematics 2014; 13: 161-170.
[11] Nabila S, Salman A. The rainbow connection number of origami graphs and pizza graphs. Procedia Computer Science 2015; 74: 162-167. doi: 10.1016/j.procs.2015.12.093
[12] Resty D, Salman A. The rainbow connection number of an n-crossed prism graph and its corona product with a trivial graph. Procedia Computer Science 2015; 74: 143-150. doi: 10.1016/j.procs.2015.12.090
[13] Simamora D, Salman A. The rainbow (vertex) connection number of pencil graphs. Procedia Computer Science 2015; 74: 138-142. doi: 10.1016/j.procs.2015.12.089

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

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