Unsolved Problems 1-25. Unsolved Problems 26-52.

10. Determine which simple graphs

Yair Caro:

- There are only seven known UPC-graphs (
*unipancyclic graphs*) having respectively 3, 5, 8, 8, 14, 14, 14 vertices (Entringer).

*n=3*, Hamilton cycle*v*._{1}v_{2}v_{3}v_{1}

*n=5*, Hamilton cycle*v*and chord_{1}v_{2}v_{3}v_{4}v_{5}v_{1}*v*._{1}v_{3}

*n=8*, Hamilton cycle*v*and_{1}v_{2}v_{3}...v_{8}v_{1}- chords
*v*and_{1}v_{3}*v*; or_{3}v_{6} - chords
*v*and_{1}v_{3}*v*._{4}v_{7}

- chords
*n=14*, Hamilton cycle*v*and_{1}v_{2}v_{3}...v_{14}v_{1}- chords
*v*,_{1}v_{3}*v*and_{1}v_{12}*v*; or_{2}v_{10} - chords
*v*,_{1}v_{3}*v*and_{2}v_{8}*v*; or_{4}v_{7} - chords
*v*,_{1}v_{3}*v*and_{2}v_{8}*v*._{5}v_{8}

- chords

- It is proved that there are only four outerplanar UPC-graphs having
3, 5, 8, 8
vertices respectively.

- It is conjectured that the above 7 graphs are the only UPC-graphs.

- Essentially there was no progress since 1986 !!

- Shi, Y.B., Yap, H.P,. and Teo, S.K.,
*On uniquely r-pancyclic graphs*,**Graph Theory and Its Applications: East and West**, (Jinan, 1986), 487-499, Ann. New York Acad. Sci., 576, New York Acad. Sci. New York, 1989. MR 93d:05088

- Shi, Y.B.,
*Some theorems of uniquely pancyclic graphs*,**Discrete Math. 59**(1986), 167-180. MR 87j:05103

Last modified December 29, 1999, by S.C. Locke.