Construction of Polytopal Graphs

Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity J. NeSeBil and M. Fiedler (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved.

Construction of PolytopaI Graphs W. SCHONE

Let be P2 = C(6,4) the cyclic 4-polytope with 6 vertices. The graph G = is the complete graph K s with the vertex set X = { 1 , 2 , . . . ,6}. We denote the facets of P2 by Fk and their vertex sets by I k where k = 1, 2, . . ., 9. We unite the sets t o a set S = {II,: II,c X ,k = 1 , 2 , . . . ,9} called the facet set of G. For k = 1, 2, . . . , 9 we name by G k the subgraph of G induced by 1,. We can assign t o G another graph G' which is the graph of a 4-polytope P;* dual t o P!. We label the vertices of P;' and G* by I I ,1 2 , . . ., Z9. Two vertices Ij Ik of G* are joined by an edge if and only if IJ f l Ik = U,, a 6 { 1 , 2 , . . . , IS} where U , is the vertex set of a subfacet F: of P:. In this case we label the edge u, = ( I j , I k ) of G" by U,. To G* belongs a facet set

(X, W ) of P;

S* = {Z;, I ; , . . . , I ; }

with I;

c S.

Fig. 1 shows the graph G' with the set S . We take a k-face with the vertex set I C X of P:. For example we choose I = {2,5}. So we have to put k = 1. Then 293

294

W. Schone

we delete from S all facets containing I and we get so the set F ( I ) of remaining facets. Further we consider the set ~ ’ ( 1=)

(u,:

It exists one and only one

Ik

E F ( I ) with U ,

c Ik}.

Now we can define an operation w 1 , I which transforms the graph G of P; in a polytopal graph G’= (X’, W’) with 7 vertices. We find X’= X U (7) and S’ = F ( I ) U { ( U , u (7)): U , E F’(I)} as a facet set of G‘. In our example we have

S’ = { { l , 2 , 3 , 4 ) , { 1 , 2 , 3 , 6 > , ( 1 , 2 , 4 , 7 ) ,{ 1 , 2 , 6 , 7 ) , { 1 , 4 , 5 , 7 ) , { 1 , 5 , 6 , 7 1 , { 1 , 4 , 5 , 6 ) ,( 2 , 3 , 6 , 7 ) , { 2 , 3 , 4 , 7 ) , { 3 , 4 , 5 , 6 ) , ( 3 , 4 , 5 , 7 ) ,( 3 , 5 , 6 , 7 ) , ( 1 , 3 , 4 , 6 ) } . The edge set W’ of G’ consists of all edges of the subgraphs Gk with V ( G k )= F ( I ) and of all edges which join the new vertex 7 with every vertex belonging to a set U , of F s ( I ) . T h e graph G’ is the graph of the simplicia1 4-polytope Pz in the notation of B. Griinbaum ([l]).We write P l = wl,1P,6. To w1,I belongs a dual operation w ; , which ~ transforms the dual graph G* of G with S = ( S ; , Sf, . . . , Sg } in the dual graph G*’ with S” = {ST’, Sf’, . . . , S;’}. T h e operation w ; , ~ can , be realized geometrically by cutting off all and only all vertices of S \ F ( I ) from the polytope P,6* by a hyperplane. All edges of P;’ which join a vertex of F ( I ) with a vertex of S \ F ( I ) are also cut by this hyperplane. We delete in G* all vertices of S \ F ( I ) and all edges joining these vertices and also all halves of cut edges incident with the deleted vertices. At the end of the remaining half of every cut edge labeled with U, we put a new vertex and denote it, by U , U (7). Two new vertices U, and Up U (7) are joined by a new edge with the label (U, U (7)) fl (Up U (7)) if and only if the sets U, U (7) and Up U (7) have three common elements. We write w ; , ~ P ; *= Pi*. Analogously to w1,I and w ; , ~we can define two other operations w1,u and w ; , ~ .Here the set = {Uik:k = 1 , 2 , .. . , r ) is a set of vertex sets of subfacets of P;. To every U,k there corresponds an edge U i k ( k = 1, 2 , . . ., r> with the label U,k in G’. The set U has the property that the subgraph G‘ of G’ generated by the edges uik ( k = 1, 2 , . . ., I-)is connected and must be a proper subgraph of the 1-skeleton of a facet of P;*. We delete from S all facets which contain an element u,k of U and we denote by F ( U ) the set of remaining facets in S . Then we put F s ( U ) = (U,: It exists one and only one I k E F ( U ) with U , C I k } . So we get a new polytopal graph G” = ( X ” , W”) with X” = X U (7) and S” = F ( U ) U { ( U , U ( 7 ) ) : U , E F s ( U ) } , where W” consists of all edges of the subgraphs Gk with Ik E F ( U ) and of all edges joining the vertex 7 with every vertex belonging to a subfacet with a vertex set U , E F’(U). T h e dual operation w : , ~can be defined by cutting off all vertices and edges of the subgraph G’ by a hyperplane from the polytope P:*. T h e operations w1,u and wT,-, cannot be realized geometrically in every case. But in the case that to U corresponds a set of edges which is a path of G* contained in the graph of a 2-face of P:* then the operations w1,u and w : , ~are geometrically realizable. This remains true for Ik E

u

Constructaon of Polytopal Graphs

295

all simplicial &polytopes. By the operations w 1 , I and w ; , ~ where , U corresponds to such a path we get all 37 combinatorial types of simplicial 4-polytopes with 8 vertices (or of simple 4-polytopes with 8 facets) from the 4-simplex. For an example we choose U = { {2,4,5}, {2,3,5}} and we get from the polytope Pt respectively p;' the sets F ( U ) = {{1,213,4},{1,2,3,6),{1,3,4,6)} and

K7

FIGURE 2 In Fig. 2 the graph G" and the graph G*" of P i and Pi* are drawn. T h e facet set of GI' is given by

The graph G is the complete graph with 7 vertices. References

[l] B. Grunbaum, V. P. Sreedharan, A n Enumeration of Simplicia1 Q-Polytopes with 8 Vertices, Journal of Combinatorial Theory 2 no. 4 (1967). [a] W. Schone, Special systems of linear equations and graphs of convex polytopes, t o be reprinted in the Ringel-Festschrift 1990.

296

w.Schone

Dr. W . Schone Technische Universitat Chemnitz, Sektion Mathematik, PSF 964, 9010 Chemnitz, Germany