Nonplanar Graphs

Theorem. Any nonplanar graph which has no automorphisms of order two is intrinsically chiral [28J. 

The chirality of the compounds discussed so far, especially of those in Section 2 is caused entirely by the chirality of their substituents. Due to their nonplanar shape, calixarenes offer various additional possibilities to produce chiral derivatives, the chirality of which is not based on a chiral group or subunit but on the absence of a symmetry plane, an inversion center, or an alternating axis in the molecule as a whole. This means, that opening of the macrocyclic structure would lead to an achiral linear molecule. Such molecules may therefore be called inherently chiral which should not be confused with the term intrinsically chiral,110 A graph whose chirality is independent of its embedding in the three-dimensional space is intrinsically chiral, while the inherent chirality defined above is due to the three-dimensional structure. 

Catenanes have a nonplanar molecular graph, which means that it will be impossible to draw their backbone on a sheet of paper without introducing crossing... 

Finally, we mention that nonplanar graphs having one unavoidable intersection of edges can be represented on the surface of a torus without such an intersection. The nonplanar graph as well as the skeleton graph of the Mobius compound synthesized by Walba et dl (which is homeomorphic to /Ca ) can be drawn on a torus surface without intersections. The surface of the torus is also important for other unusual chemical compounds The skeletons of catenanes, rotaxanes and knots cannot be embedded in the surface of a three-dimensional sphere, but in that of a torus. 

Figure 4. (a) A classical rotaxane (b) a rotaxane with an infinitely long thread (c) a catenane. Note that (b) and (c) have nonplanar molecular graphs. 

Not surprisingly, the CgHg isomer with maximal steric energy (688.76 kcal/mol) corresponds to the nonplanar graph kTj 3. This structure is shown as 3, followed by the isomers of next-highest steric energy, the tetrahedrenes 4 and 5 ... 

Benzenoids and single coronoids represent two main classes of polyhexes (Sect. 2.1). The inclusion of multiple coronoids represents an extension in one direction. Classes of multiple coronoids are defined and discussed in the next chapter. Here we shall concentrate upon polyhexes without holes (as benzenoids) and those dth one hole each (as single coronoids) and discuss an extension which involves geometrically nonplanar (helicenic) systems. We shall find it expedient to define a class where the members are restricted to helicenic systems which can be represented by planar graphs. In the next section, polyhex systems which cannot be represented by planar graphs, are exemplified. 

Definition 2.5 A corohelicene (system) is a multiply connected, geometrically nonplanar (helicenic) polyhex, which is graph-theoretically planar. 

It remains to treat graph-theoretically nonplanar polyhexes. We shall not attempt to give any systematic classification of the variety of these systems, but present some examples in the next section. 

Fig. 2.6. An isomer of C gH24 [12]cyclohelicene the dualist representation (left) and a nonplanar graph (right).

Here the left-hand system is graph—theoretically nonplanar like the systems of Figs. 6 and 7. However, the right-hand isomer is a graph—theoretically planar system and actually a corohelicene similar to the system of Fig. 4. 

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