J Austral Math Soc Ser A 53 pp402--407, 1992.

Of Zero-Sum Turan Problems of Bialstocki and Dierker

Y. Caro and Y. Roditty

Abstract

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 Turan 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, tK₂, Z_k) for k | t. In particular, is it true that T(n, tK₂, Z_k) = T(n, (t + k - 1)K₂)? 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 AMS Subject Classification: 05C55

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Authors

Y. Caro

School of Education, University of Haifa-ORANIM, Tivon, ISRAEL 36910.

Y. Roditty

Department of Mathematics, Beit-Berl College, Kfar-Saba, ISRAEL 44905.