Sparseness of 4-Cycle Systems

An avoidance problem of configurations in \( 4 \)-cycle systems is investigated by generalizing the notion of sparseness, which is originally from Erdős’ \( r \)-sparse conjecture on Steiner triple systems. A \( 4 \)-cycle system of order \( v \), \( 4CS(v) \), is said to be \( r \)-sparse if for every integer \( j \) satisfying \( 2 \leq j \leq r \) it contains no configurations consisting of \( j \) \( 4 \)-cycles whose union contains precisely \( j + 3 \) vertices. If an \( r \)-sparse \( 4CS(v) \) is also free from copies of a configuration on two \( 4 \)-cycles sharing a diagonal, called the double-diamond, we say it is strictly \( r \)-sparse. In this paper, we show that for every admissible order \( v \) there exists a strictly \( 4 \)-sparse \( 4CS(v) \). We also prove that for any positive integer \( r \geq 2 \) and sufficiently large integer \( v \), there exists a constant number \( c \) such that there exists a strictly \( r \)-sparse \( 4 \)-cycle packing of order \( v \) with \( c \cdot v^2 \) \( 4 \)-cycles.