Abstract
A copy of a graph H in an edge colored graph G is called rainbow if all edges of H have distinct colors. The size anti-Ramsey number of H, denoted by \(AR_s(H)\), is the smallest number of edges in a graph G such that any of its proper edge-colorings contains a rainbow copy of H. We show that \(AR_s(K_k) = \varTheta (k^6 / \log ^2 k)\). This settles a problem of Axenovich, Knauer, Stumpp and Ueckerdt. The proof is probabilistic and suggests the investigation of a related notion, which we call the degree anti-Ramsey number of a graph.
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Acknowledgments
I would like to thank Maria Axenovich for telling me about the problem suggested in [3] and for helpful discussions. Part of this work was done during the Japan Conference on Graph Theory and Combinatorics, which took place in Nihon University, Tokyo, in May, 2014. I would like to thank the organizers of the conference for their hospitality. Research supported in part by a USA-Israeli BSF Grant, by an ISF Grant, by the Israeli I-Core program and by the Oswald Veblen Fund.
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Alon, N. Size and Degree Anti-Ramsey Numbers. Graphs and Combinatorics 31, 1833–1839 (2015). https://doi.org/10.1007/s00373-015-1583-9
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DOI: https://doi.org/10.1007/s00373-015-1583-9