On Zero-Sum Turan Numbers: Stars and Cycles

Abstract

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 $\sum _{e\in E(G)}c(e)\equiv 0(\mathrm{mod}k)$. 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 $m\ge k\ge 2$ be integers, $k|m$. Suppose $n>2(m-1)(k-1)$, then
$$T(n,K_{1,m},Z_k)=\{\begin{array}{ll}\frac{(m+k-2)-n}{2}-1,& ifn-1\equiv m\equiv k\equiv 0(\mathrm{mod}2);\\ \lfloor \frac{(m+k-2)-n}{2}\rfloor ,& otherwise.\end{array}$$