A note on traversing specified vertices in graphs embedded with large representativity

A graph G is said to have property P ( 2 , k ) if given any k + 2 distinct vertices a , b , v 1 , … , v k , there is a path P in G joining a and b and passing through all of v 1 , … , v k . A graph G is said to have property C ( k ) if given any k distinct vertices v 1 , … , v k , there is a cycle C in G containing all of v 1 , … , v k . It is shown that if a 4-connected graph G is embedded in an orientable surface Σ (other than the sphere) of Euler genus e g ( G , Σ ) , with sufficiently large representativity (as a function of both e g ( G , Σ ) and k ), then G possesses both properties P ( 2 , k ) and C ( k ) .