[vc_empty_space][vc_empty_space]
The strong 3-rainbow index of comb product of a tree and a connected graph
Awanis Z.Y.a, Salman A.N.M.a, Saputro S.W.a
a Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, 40132, Indonesia
[vc_row][vc_column][vc_row_inner][vc_column_inner][vc_separator css=”.vc_custom_1624529070653{padding-top: 30px !important;padding-bottom: 30px !important;}”][/vc_column_inner][/vc_row_inner][vc_row_inner layout=”boxed”][vc_column_inner width=”3/4″ css=”.vc_custom_1624695412187{border-right-width: 1px !important;border-right-color: #dddddd !important;border-right-style: solid !important;border-radius: 1px !important;}”][vc_empty_space][megatron_heading title=”Abstract” size=”size-sm” text_align=”text-left”][vc_column_text]© 2020 Information Processing Society of Japan.Let G be a nontrivial connected graph of order n. Let k be an integer with 2 ≤ k ≤ n. A strong k-rainbow coloring of G is an edge-coloring of G having property that for every set S of k vertices of G, there exists a tree with minimum size containing S whose all edges have distinct colors. The minimum number of colors required such that G admits a strong k-rainbow coloring is called the strong k-rainbow index srxk (G) of G. In this paper, we study the strong 3-rainbow index of comb product between a tree and a connected graph, denoted by Tn ⊲o H. Notice that the size of Tn ⊲o H is the trivial upper bound for srx3 (Tn ⊲o H), which means we can assign distinct colors to all edges of Tn ⊲o H. However, there are some connected graphs H such that some edges of Tn ⊲o H may be colored the same. Therefore, in this paper, we characterize connected graphs H with srx3 (Tn ⊲o H) = |E(Tn ⊲o H)|. We also provide a sharp upper bound for srx3 (Tn ⊲o H) where srx3 (Tn ⊲o H) ≠ |E(Tn ⊲o H)|. In addition, we determine the srx3 (Tn ⊲o H) for some connected graphs H.[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Author keywords” size=”size-sm” text_align=”text-left”][vc_column_text][/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Indexed keywords” size=”size-sm” text_align=”text-left”][vc_column_text]Comb product,Rainbow coloring,Strong k-rainbow index[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Funding details” size=”size-sm” text_align=”text-left”][vc_column_text]Acknowledgments This research was supported by a research grant from Ministry of Research, Technology and Higher Education, Indonesia.[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”DOI” size=”size-sm” text_align=”text-left”][vc_column_text]https://doi.org/10.2197/ipsjjip.28.865[/vc_column_text][/vc_column_inner][vc_column_inner width=”1/4″][vc_column_text]Widget Plumx[/vc_column_text][/vc_column_inner][/vc_row_inner][/vc_column][/vc_row][vc_row][vc_column][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][/vc_column][/vc_row]