# 线性代数网课代修|ENGO361代写|ENGO361英文辅导|ENGO361

# Directed graphs without rainbow stars+ The work of the first author was supported by the National Research, Development and Innovation Office – NKFIH under the grants FK 132060 and KKP-133819. The work of the second and the fourth author was supported by the National Science Centre grant 2021/42/E/ST1/00193. The work of the third author was supported by a grant from the Simons Foundation #712036.Daniel GerbnerAlfred Renyi Institute of Mathematics, HUN-REN. E-mail: gerbner.daniel@renyi.hu.Andrzej GrzesikFaculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Krakow, Poland. E-mail: Andrzej.Grzesik@uj.edu.pl.Cory PalmerDepartment of Mathematical Sciences, University of Montana. E-mail: Cory.palmer@umontana.edu.Magdalena ProrokAGH University of Krakow, al. Mickiewicza 30, 30-059 Krakow, Poland. E-mail: prorok@agh.edu.pl.Such a rainbow Turan problem was also considered for directed graphs in [3], where the optimal solution (up to a lower order error term) for $\min_{1\leq i\leq c}e(G_{i})$ and $\sum_{1\leq i\leq c}e(G_{i})$ was provided, for any number of colors, when a directed or transitive rainbow triangle is forbidden. Here, we continue this line of research on directed graphs and consider a directed star as the forbidden rainbow graph.Let $S_{p,q}$ be the orientation of a star on $p+q+1$ vertices with _center_ vertex of indegree $p$ and outdegree $q$. Forbidding a rainbow $S_{p,q}$ in a collection of graphs $\mathcal{G}=(G_{1},\ldots,G_{c})$ is analogous to forbidding a rainbow $S_{q,p}$ in the collection of graphs obtained by changing the orientation of every edge in each graph from $\mathcal{G}$. Thus it is enough to consider this rainbow Turan problem for $S_{p,q}$ only when $p\leq q$. As this problem is trivial when the number of colors $c$ is less than the number of edges in the forbidden rainbow graph, we consider only $c\geq p+q$.In Section 2 we consider a star $S_{0,q}$ as the forbidden rainbow graph and prove, for every $n>c\geq q\geq 1$, exact bounds for $\sum_{i=1}^{c}e(G_{i})$ () and $\min_{1\leq i\leq c}e(G_{i})$ (). In Section 3 we consider $S_{p,q}$ as the forbidden rainbow graph for any $q\geq p\geq 1$ and prove bounds for $\sum_{i=1}^{c}e(G_{i})$ () and $\min_{1\leq i\leq c}e(G_{i})$ (), which are tight up to a lower order error term. Additionally, in Section 4 we provide exact bounds for any $c\geq 2$ and $n\geq 3$ when a rainbow $S_{1,1}$, i.e., directed path of length $2$, is forbidden, for both the sum () and the minimum () of the number of edges.