Abstract
In an edge-colored graph G, a rainbow clique Kk is a complete subgraph on k vertices in which all the edges have distinct colors. Let e(G) and c(G) be the number of edges and colors in G, respectively. In this paper, we show that for any ɛ>0, if e(G)+c(G)≥(1+k−3k−2+2ɛ)n2 and k≥3, then for sufficiently large n, the number of rainbow cliques Kk in G is Ω(nk).We also characterize the extremal graphs G without a rainbow clique Kk, for k=4,5, when e(G)+c(G) is maximum.Our results not only address existing questions but also complete the findings of Ehard and Mohr (2020).
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