On zero-sum turan problems of Bialostocki and Dierker

Y. Caro, Y. Roditty
1992 Journal of the Australian Mathematical Society  
Assume G is a graph with m edges. By T(n, G) we denote the classical Turan number, namely, the maximum possible number of edges in a graph H on n vertices without a copy of G . Similarly if G is a family of graphs then H does not have a copy of any member of the family. A Z k -colouring of a graph G is a colouring of the edges of G by Z k , the additive group of integers modulo k , avoiding a copy of a given graph H, for which the sum of the values on its edges is 0 (mod k). By the Zero-Sum
more » ... n number, denoted T(n, G, Z k ), k | m , we mean the maximum number of edges in a Z k -colouring of a graph on n vertices that contains no zero-sum (mod k) copy of G . Here we mainly solve two problems of Bialostocki and Dierker [6]. PROBLEM 1. Determine T(n , tK 2 , Z k ) for k \ t. In particular, is it true that T(n , tK 2 , Z k ) = T(n,(t + k-1)# 2 )? PROBLEM 2. Does there exist a constant c(t, k) such that T{n , F t , Z k ) <c(t, k)n , where F t is the family of cycles of length at least t ? 1991 Mathematics subject classification (Amer. Math. Soc): 05 C 55.
doi:10.1017/s1446788700036569 fatcat:g3jhwm3lu5cqtgjodik4dgmmuu