Multi-cyclic graphs in the random graph process with restricted budget

We study a controlled random graph process introduced by Frieze, Krivelevich, and Michaeli. In this model, the edges of a complete graph are randomly ordered and revealed sequentially to a builder. For each edge revealed, the builder must irrevocably decide whether to purchase it. The process is subject to two constraints: the number of observed edges $t$ and the builder's budget $b$. The goal of the builder is to construct, with high probability, a graph possessing a desired property. Previously, the optimal dependencies of the budget $b$ on $n$ and $t$ were established for constructing a graph containing a fixed tree or cycle, and the authors claimed that their proof could be extended to any unicyclic graph. The problem, however, remained open for graphs containing at least two cycles, the smallest of which is the graph $K_4^-$ (a clique of size four with one edge removed). In this paper, we provide a strategy to construct a copy of the graph $K_4^-$ if $b \gg \max\left\{n^6 / t^4, n^{4 / 3} / t^{2 / 3}\right\}$, and show that this bound is tight, answering the question posed by Frieze et al. concerning this specific graph. We also give a strategy to construct a copy of a graph consisting of $k$ triangles intersecting at a single vertex (the $k$-fan) if $b \gg \max\left\{n^{4k - 1} / t^{3k - 1}, n / \sqrt{t}\right\}$, and also show that this bound is tight. These are the first optimal strategies for constructing a multi-cyclic graph in this random graph model.