On Zero-Sum Turan Numbers: Stars and Cycles

Let \( {Z}_k\) be the cyclic group of order \( k\). Let \( H\) be a graph. A function \( c: E(H) \to {Z}_k\) is called a \( {Z}_k\)-coloring of the edge set \( E(H)\) of \(H\). A subgraph \( G \subseteq H\) is called zero-sum (with respect to a \( {Z}_k\)-coloring) if \( \sum_{e \in E(G)} c(e) \equiv 0 \pmod{k}\). Define the zero-sum Turán numbers as follows. \( T(n, G, {Z}_k)\) is the maximum number of edges in a \( {Z}_k\)-colored graph on \( n\) vertices, not containing a zero-sum copy of \( G\). Extending a result of [BCR], we prove:

THEOREM.
Let \( m \geq k \geq 2\) be integers, \( k | m\). Suppose \( n > 2(m-1)(k-1)\), then
\[T(n,K_{1,m},{Z}_k) =
\left\{
\begin{array}{ll}
\frac{(m+k-2)-n}{2}-1, & if \;\; n-1 \equiv m \equiv k \equiv 0 \pmod{2}; \\
\lfloor \frac{(m+k-2)-n}{2} \rfloor, & otherwise.
\end{array}
\right.\]