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2 Rainbow directed \(S_{0,q}\)In this section the forbidden rainbow graph is a directed star with all edges oriented away from the center. As noted earlier, the problem is the same if we forbid a directed star with all edges oriented to the center. The following theorem provides the optimal bound for the sum of the number of edges in all graphs.**Theorem**.: _For integers \(n>c\geq q\geq 1\), every collection of directed graphs \(G_{1},\ldots,G_{c}\) on a common set of \(n\) vertices containing no rainbow \(S_{0,q}\) satisfies_\[\sum_{i=1}^{c}e(G_{i})\leq(q-1)(n^{2}-n).\]_Moreover, this bound is sharp._Proof.: The sharpness of the bound follows from taking a collection of \(q-1\) complete graphs, but there are more extremal constructions. We split the vertex set into disjoint subsets and _assign_ to each of them a different set of \(q-1\) colors. This way each vertex has \(q-1\) colors assigned. Then, for any two vertices \(u\), \(v\) and color \(i\in[c]\), we add edge \(uv\) to \(G_{i}\) if color \(i\) is assigned to \(u\). Clearly this construction does not contain a rainbow \(S_{0,q}\) as no vertex has positive outdegree in \(q\) graphs, and each vertex has the sum of outdegrees over all graphs equal to \((q-1)(n-1)\) giving the total number of edges equal to \((q-1)(n^{2}-n)\). An example construction of this type is shown in To prove the upper bound, consider a collection of graphs \(G_{1},G_{2},\ldots,G_{c}\) on a common set \(V\) of \(n\) vertices that does not contain a rainbow \(S_{0,q}\). Let \(v\in V\) be an arbitrary vertex. We will show that the total number of edges outgoing from \(v\) is bounded by \((q-1)(n-1)\)by connecting vertex \(u\in V\setminus\{v\}\) and color \(i\in[c]\) if \(vu\in E(G_{i})\). The number of edges in \(H\) is equal to the number of outgoing edges from \(v\) that we want to bound. Note that the existence of a matching of size \(q\) in \(H\) means that there exists a rainbow \(S_{0,q}\) with center \(v\), which is not possible. Therefore, the maximum matching in \(H\) is of size at most \(q-1\). From Konig’s theorem this means that the minimum vertex cover is also of size at most \(q-1\). As \(n-1\geq c\), this implies that the maximum number of edges in \(H\) is bounded by \((q-1)(n-1)\), as desired. By summing it over all vertices we obtain \(\sum_{i=1}^{c}e(G_{i})\leq(q-1)(n^{2}-n)\). Note that the bound \(n>c\) in is indeed needed, as otherwise the following collections of graphs contradict the theorem. If \(c\geq n\geq q\), for each vertex \(v\), add edges in each color from \(v\) to \(q-1\) other arbitrary vertices. This way\[\sum_{i=1}^{c}e(G_{i})=(q-1)cn>(q-1)(n^{2}-n)\]and there is no rainbow \(S_{0,q}\). While if \(n\leq q\), then a collection of complete directed graphs does not contain rainbow \(S_{0,q}\) and satisfies\[\sum_{i=1}^{c}e(G_{i})=c(n^{2}-n)>(q-1)(n^{2}-n).\]The same constructions also provide an exact bound for \(\min_{1\leq i\leq c}e(G_{i})\) for any \(n\leq c\). implies that for integers \(n>c\geq q\geq 1\), every collection of directed graphs \(G_{1},\ldots,G_{c}\) on a common set of \(n\) vertices containing no rainbow \(S_{0,q}\) satisfies\[\min_{1\leq i\leq c}e(G_{i})\leq\frac{q-1}{c}\left(n^{2}-n\right).\]Moreover, if \(n(q-1)\) is divisible by \(c\), then one can make a construction as detailed in the beginning of the proof of , in which the number of edges in each graph is the same (see **Theorem**.: _For integers \(n>c\geq q\geq 1\), every collection of directed graphs \(G_{1},\ldots,G_{c}\) on a common set of \(n\) vertices containing no rainbow \(S_{0,q}\) satisfies_\[\min_{1\leq i\leq c}e(G_{i})\leq\left\lfloor\frac{n(q-1)}{c}\right\rfloor(n-1)+r,\]_where \(r\) is the remainder of \(n(q-1)\) when divided by \(c\). Moreover, this bound is sharp._Proof.: We proceed similarly as in the proof of . Consider a collection of graphs \(G_{1},G_{2},\ldots,G_{c}\) on a common set \(V\) of \(n\) vertices that does not contain a rainbow \(S_{0,q}\). For any vertex \(v\in V\) we consider an auxiliary bipartite graph \(H\) between the vertices \(V\setminus\{v\}\) and colors in \([c]\) created by connecting vertex \(u\in V\setminus\{v\}\) and color \(i\in[c]\) if \(vu\in E(G_{i})\). Since the existence of a matching of size \(q\) in \(H\) means that there exists a forbidden rainbow \(S_{0,q}\) centered in \(v\), the maximum matching in \(H\) is of size \(q-1\). From Konig’s theorem the minimum vertex cover is also of size at most \(q-1\). Let \(a_{v}\) and \(b_{v}\) be the numbers of vertices in the minimum vertex cover that are in the part of \(H\) related with the colors and in the part related with the other vertices, respectively. In particular, in \(a_{v}\) colors there are at most \(n-1\) edges outgoing from \(v\) and in all other colors there are at most \(b_{v}\) edges outgoing from \(v\). Let \(a=\sum_{v\in V}a_{v}\) and \(b=\sum_{v\in V}b_{v}\). Since \(a_{v}+b_{v}\leq q-1\), we have \(a+b\leq n(q-1)\)图片描述

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