Abstract
Colourful connection concepts in graph theory such as rainbow connection, proper connection, odd connection or conflict-free connection have received a lot of attention. For an integer \(k \ge 1\) we call a path P in a graph G k-colourful, if at least k vertices of P are pairwise differently coloured. A graph G is k-colourful connected, if any two vertices of G are connected by a k-colouful path. Now we call the least integer k, which makes G k-colourful connected, the k-colourful connection number of G. In this paper, we introduce the (strong, internal) k-colourful connection number of a graph, establish bounds for our new invariants in several graph classes as well as compute exact values for \(k\in [3]\).
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Acknowledgements
The research of all three authors was supported in part by the DAAD-PPP project “Colourings and connection in graphs” with project-ID 57210296. The work of Stanislav Jendrol’was supported by the Slovak Research and Development Agency under the contract No. APVV-19-0153 and by the Slovak VEGA Grant 1/0574/21.
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Dedicated to Professors Eiichi Bannai and Hikoe Enomoto on the occasion of their 75th birthday.
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Brause, C., Jendrol’, S. & Schiermeyer, I. From Colourful to Rainbow Paths in Graphs: Colouring the Vertices. Graphs and Combinatorics 37, 1823–1839 (2021). https://doi.org/10.1007/s00373-021-02322-9
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DOI: https://doi.org/10.1007/s00373-021-02322-9