(PI") each adjacent to one of the vertices v3 and v4,forming a hexagon. Subcases (Pi) a n d (P,")immediately lead to the multigraphs 3 a n d 1,respectively. By this t h e case (PI) is exhausted. Similar arguments show t h a t t h e only multigraph satisfying (Pz) i s 2. T h e examination of t h e connected cubic multigraphs without double edges i s analogous and results in 4 a n d 6. We, thus, establish that, i n addition to t h e five cubic multigraphs corresponding to (CH)risomers 1-5 from Figure 1,one more multimaph can be wnstructed t h a t would correspond to a hydro&&on species, 13.
valent (two-centric) bonds. that all carbon atoms are formally four-valent, and that a hydrogen atom is attached to each carbon. In other words. formula 13 reoresents a (CHIisomer obeying all t h e requirements of thcclassical t h e o j of chemical structure. The unusual feature of hydrocarbon 13 is i t s (clearly very long) C-C bond, which, of necessity, passes through a four-membered ring. If i t is possible for molecules of type 13 to exist at all, they will possess a n enormous steric strain and are expected to be extremely reactive moieties. Nevertheless, we wish to emphasize t h a t t h e set of ( C l l ~ ~ i s o m epresented rs in Figure 1 is not complete. It 1s possible to construct one more CH16-species that obeys the rules of classical structural chemistrv. Althoueh it is more ~~~t h a n likely t h a t a compound of the >ormula i 3 would be h i ~ b l yunstable and not a reasonable tareet of chemical synthesis, t h e possibility of its existence(perhaps a s a transient intermediate) should not be discounted. ~
~
Literature Cited 1. Wheland. G. WResomncp in Organic Chemistry; Wile?: New York,1955. 2. Pauling, L TheNolureoffha ChemimlBond; Cornell Univ Rees: Ithaca,New York, 1960.
A. T.Balaban seems to be t h e fust to point out this sixth v a l e n c e i s o m e r of b e n z e n e (19). H e d u b b e d i t "benzmobiusstripane". Eventually t h i s hypothetical hydrocarbon was considered by R. R. Karl and S. H. Bauer (20)and quite recently by S. D. Warren and B. M. Gimarc (21). . .
It i s by no means easy to deduce t h e actual form of t h e hydrocarbon 13. We depicted i t (somewhat arbitrarily) as a distorted octahedron. What we deem to be important i s t h a t all the atoms i n hydrocarbon 13 a r e wnnected by w -
12. Potgleter. J:H. J. Cham Educ. l'k1,68,28&281. 13. See; for instanstane,Angow C 6 m . Inf. Ed. Engl. 1889,28,1\2%9. 14. Seybold, P. 0.; May, M.; Bagal, U. A. J Chem. Educ. 1881,64,57.%581. 15. Balaban, A. T.; Kennedy,J. W: Buintas, L.V.J C 6 m Educ. 1988,66,304413. 16. Haneen,P J.; Jun, P. C. J. C6m.Edue. 1988.65.574-580, 17. Trhajstic, N. Chemical Gmph Theory. 2nd ed.: CRC Press: B- Raton, FL, 1992. 18. See the September 1992 issue of his Journal, that contains a series ofsmcles devoted to chemical applications ofgraphs: J Chem. Educ. 1992.69.690-718. 19. Balaban, A. T. Re". RoumoineChim. lW. 11, 1097-1116. 20. Kar1.R. R.;Bauer,S. H. J Mal. Sfrucl. 1975.25, 1-15. 21. Warren, D.S.; Gim8rc.B.M. J A m . Chm. Soc. 1982,114,5378-5385.
Research Associateship Opportunities The National Research Council announces the 1994Resident, Cwperative, and Postdoctoral Associateship Programs to he conducted on behalf of Federal agencies or research institutions whose 140 participating research laboratories are located throughout the United States. The programs provide opportunities for PhD scientists and engineers of unusual promise and ability to perform research on problems largely of their own chwsing yet compatible with the research interests of the sponsoring laboratory. Initiated in 1954,the Associateship programs have been awarded t o over 7000 scientists ranging from recent PhD recipients to distinguished senior scientists. Approximately 350 new full-time Associateships will he awarded this year in a broad spectrum of sciences including chemistry, earth and atmospheric sciences, and biological sciences. Most programs are open to hoth US and non-US nationals and to hoth recent PhD degree recipients and senior investigators. Awards are for one or two years, renewable to a maximum of three years. Senior applicants who have held the doctorate for at least five years may request a shorter period. Annual stipends for recent PbD's are in the $30,00045,000 range, with appropriate adjustment for senior Associates. Applications submitted directly to the National Research Council are accepted on a continuous basis. Those postmarked no later than April 15 will be reviewed in June with announcements made in July; those received by August 15 will be reviewed in October and recipients will be announced in November. Information an specific opportunites and participating laboratories, as well as application materials, may he ohtained from: Assaeiateship Programs (TJ2094iD31,National Research Council,2101 Constitution Avenue, N.W., Wasbiogtan, DC 20418.FAX: (202)334-2759,
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Journal of Chemical Education
