Sunil Chandran, L and Issac, D and Lauri, J and van Leeuwen, EJ (2022) Upper bounding rainbow connection number by forest number. In: Discrete Mathematics, 345 (7).
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Abstract
A path in an edge-colored graph is rainbow if no two edges of it are colored the same, and the graph is rainbow-connected if there is a rainbow path between each pair of its vertices. The minimum number of colors needed to rainbow-connect a graph G is the rainbow connection number of G, denoted by rc(G). A simple way to rainbow-connect a graph G is to color the edges of a spanning tree with distinct colors and then re-use any of these colors to color the remaining edges of G. This proves that rc(G)â�¤|V(G)|â��1. We ask whether there is a stronger connection between tree-like structures and rainbow coloring than that is implied by the above trivial argument. For instance, is it possible to find an upper bound of t(G)â��1 for rc(G), where t(G) is the number of vertices in the largest induced tree of G? The answer turns out to be negative, as there are counter-examples that show that even câ� t(G) is not an upper bound for rc(G) for any given constant c. In this work we show that if we consider the forest number f(G), the number of vertices in a maximum induced forest of G, instead of t(G), then surprisingly we do get an upper bound. More specifically, we prove that rc(G)â�¤f(G)+2. Our result indicates a stronger connection between rainbow connection and tree-like structures than that was suggested by the simple spanning tree based upper bound. © 2022 Elsevier B.V.
| Item Type: | Journal Article |
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| Publication: | Discrete Mathematics |
| Publisher: | Elsevier B.V. |
| Additional Information: | The copyright for this article belongs to Authors |
| Department/Centre: | Division of Electrical Sciences > Computer Science & Automation |
| Date Deposited: | 21 Mar 2022 07:35 |
| Last Modified: | 21 Mar 2022 07:35 |
| URI: | http://eprints.iisc.ac.in/id/eprint/71564 |
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