Filling of a given boundary by p-gons and related problems

We consider here (p,s)(p,s)-polycycles   (3⩽p⩽s3⩽p⩽s) i.e. plane graphs, such that all interior faces are p-gons, all interior vertices are s-valent and any vertex of the boundary   (i.e. the exterior face) has valency within [2,s][2,s]. The boundary sequence   of a (p,s)(p,s)-polycycle P   is the sequence b(P)b(P) enumerating, up to a cyclic shift or reversal, the consecutive valencies of vertices of the boundary. We show that the values p=3,4p=3,4 are the only ones, such that the boundary sequence defines its (p,3)(p,3)-filling   (i.e. a (p,3)(p,3)-polycycle with given boundary) uniquely.Also we give new results in the enumeration of maps Mn(p,q)Mn(p,q) (i.e. plane 3-valent maps with only p- and q-gonal faces, such that the q-gons are organized in an n-ring) and two of their generalizations.Both problems are similar (3-valent filling by p-gons of a boundary or of a ring of q-gons) and the same programs were used for both computations.