Cycles Through Prescribed and Forbidden Point Sets

A graph G has property C ( m + , n − ) if for any choice of m + n points u 1 ,…, u m , v 1 ,…, v n in G there is a cycle in G which includes all of u 1 ,…, u m , but none of v 1 ,…, v n . We discuss the family of implications ‘ C ( m + , n − )→ C ( r + , s − )’ for various non-negative integral values of m , n , r and s . The general question as to when such implications hold seems quite difficult. We discuss some reductions of these problems and prove the implication C ( n + , 1 − )→ C ( n + 1 + , 0 − ) valid for n = 2, 3, and 4. The Petersen graph shows that this implication fails for n = 9.