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

Abstract

We consider here ( p , s ) - polycycles ( 3 ⩽ 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 ] . The boundary sequence of a ( p , s ) -polycycle P is the sequence 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 , 4 are the only ones, such that the boundary sequence defines its ( p , 3 ) - filling (i.e. a ( p , 3 ) -polycycle with given boundary) uniquely. Also we give new results in the enumeration of maps M n ( 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.