Vertex-vectors of quadrangular 3-polytopes with two types of edges.

*(English)*Zbl 0705.05043
Combinatorics and graph theory, Proc. 30th Semester, Warsaw/Pol. 1987, Banach Cent. Publ. 25, 93-111 (1989).

[For the entire collection see Zbl 0699.00015.]

An edge of a graph is said to be of type (a,b) if it connects vertices with degree a and b. Consider the family of all 3-dimensional polytopes all faces of which are quadrangles and all edges of which are of one of two types only. Let S(a,b,c) be the family of the edge-graphs of such polytopes with edges of types (a,b) and (b,c). It is known for which values (a,b,c), S(a,b,c) is finite (nonvoid) or infinite.

This paper addresses the problem for which values (va,vb,vc) some graph in such an S(a,b,c) exists with exactly vi vertices of degree i \((i=a,b,c)\). Many existence and non-existence results are proved, but several cases remain undecided.

Most existence results are obtained by way of the radial construction: for a planar graph first select an inner point of each face, and connect it to each vertex of that face, and finally delete all original edges. If the starting graph is 2-vertex connected and 3-edge connected with all vertices of degree b and all faces either a-gons or c-gons, then this construction yields a graph of S(a,b,c).

An edge of a graph is said to be of type (a,b) if it connects vertices with degree a and b. Consider the family of all 3-dimensional polytopes all faces of which are quadrangles and all edges of which are of one of two types only. Let S(a,b,c) be the family of the edge-graphs of such polytopes with edges of types (a,b) and (b,c). It is known for which values (a,b,c), S(a,b,c) is finite (nonvoid) or infinite.

This paper addresses the problem for which values (va,vb,vc) some graph in such an S(a,b,c) exists with exactly vi vertices of degree i \((i=a,b,c)\). Many existence and non-existence results are proved, but several cases remain undecided.

Most existence results are obtained by way of the radial construction: for a planar graph first select an inner point of each face, and connect it to each vertex of that face, and finally delete all original edges. If the starting graph is 2-vertex connected and 3-edge connected with all vertices of degree b and all faces either a-gons or c-gons, then this construction yields a graph of S(a,b,c).

Reviewer: F.Plastria