On the divisibility of the cycle number by 7 (Q687147)

It is shown that for every even \(p \geq 4\), there exists a nonseparable cubic graph \(G\), such that the number of cycles in \(G\) is a multiple of seven. This answers a question by Gerhard Ringel. It is also shown that there exists a nonseparable cubic graph whose cycle number is congruent to \(k \pmod 7\) where \(k=0\) when \(p \geq 4\), \(k=1\) when \(p \geq 6\), \(k=5\) when \(p \geq 8\), and \(k=2,3,4,6\) when \(p \geq 10\).