Abstract
This paper explores a new direction in the well studied area of graph saturation—we introduce the notion of rainbow saturation. A graph G is rainbow H-saturated if there is some proper edge coloring of G which is rainbow H-free (that is, it has no copy of H whose edges are all colored distinctly), but where the addition of any edge makes such a rainbow H-free coloring impossible. Taking the maximum number of edges in a rainbow H-saturated graph recovers the rainbow Turán numbers whose systematic study was begun by Keevash, Mubayi, Sudakov, and Verstraëte. In this paper, we initiate the study of corresponding rainbow saturation number—the minimum number of edges among all rainbow H-saturated graphs. We give examples of graphs for which the rainbow saturation number is bounded away from the traditional saturation number (including all complete graphs \(K_n\) for \(n\ge 4\) and several bipartite graphs). It is notable that there are non-bipartite graphs for which this is the case, as this does not happen when it comes to the rainbow Turán number versus the traditional Turán number. We also show that saturation numbers are at most linear for a large class of graphs, providing a partial rainbow analogue of a well known theorem of Kászonyi and Tuza. We conclude this paper with a number of related open questions and conjectures.
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Notes
Note that we use \(P_k\) to denote the path with k vertices and \((k-1)\) edges. Further, note that since every proper coloring of \(P_3\) is rainbow, we know that \({{\text {sat}}^\star }(n,P_3)={{\text {sat}}}(n,P_3)\).
That is, H contains no induced even cycle
We note that adding edges in a different order may result in a different graph; any such graph will suffice.
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Bushaw, N., Johnston, D. & Rombach, P. Rainbow Saturation. Graphs and Combinatorics 38, 166 (2022). https://doi.org/10.1007/s00373-022-02566-z
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DOI: https://doi.org/10.1007/s00373-022-02566-z