Pacific Journal of Mathematics

Paths on polyhedra. II.

Victor Klee

Article information

Source
Pacific J. Math., Volume 17, Number 2 (1966), 249-262.

Dates
First available in Project Euclid: 8 December 2004

Permanent link to this document
https://projecteuclid.org/euclid.pjm/1102994619

Mathematical Reviews number (MathSciNet)
MR0194982

Zentralblatt MATH identifier
0141.21303

Subjects
Primary: 52.10

Citation

Klee, Victor. Paths on polyhedra. II. Pacific J. Math. 17 (1966), no. 2, 249--262. https://projecteuclid.org/euclid.pjm/1102994619


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References

  • [1] P. Alexandroff and H. Hopf, Topologie I, Springer, Berlin, 1935.
  • [2] T. A. Brown, Simple paths on convex polyhedra, Pacific J. Math. 11 (1961), 1211- 1214.
  • [3] M.Bruckner, ber die Ableitung der allgemeinen Polytope und die nach Isomorphismus verschiedenen Typen der allgemeinen Achtzelle (Oktatope), Verhandelingen der Konink- lijke Akademie van Wetenschappen te Amsterdam (Eerste Sectie), Deel X, No. 1 (1909), 29 pages -f 2 plates.
  • [4] C. Caratheodory, ber den Variabilitatsbereich der Fourierschen Konstantenvon positiven harmonischen Funktionen,Rend. Circ. Mat. Palermo, 32 (1911), 193-217.
  • [5] G.B Dantzig, Linear programming and extensions, Princeton Univ. Press, Prince- ton, N. J., 1963.
  • [6] D. Gale, Neighborly and cyclic polytopes, Proc. Symp. Pure. Math., 7, Convexity, Amer. Math. Soc, Providence, R.I., (1963), 225-232.
  • [7] D. Gale, On the number of faces of a convex polytope, Canad. J. Math. 16 (1964), 12-17.
  • [8] B. Grnbaum, Convex polytopes, Wiley, N. Y., 1966.
  • [9] B. Grnbaum and T. S. Motzkin, Longest simple paths in polyhedral graphs, J. London Math. Soc. 37 (1962), 152-160.
  • [10] V. Klee, Diameters of polyhedral graphs, Canad. J. Math. 16 (1964), 602-614.
  • [11] V. Klee, On the number of vertices of a convex polytope, Canad. J. Math. 16 (1964), 701-720.
  • [12] V. Klee, Heights of convex polytope, J. Math. Anal. Appl., 11 (1965), 176-190.
  • [13] Convex polytopes and linear programming, Proceedings of the IBM Scientific Com- puting Symposium on Combinatorial Problems (March 16-18, 1964), 1966.
  • [14] Convex polytopes and linear programming, Paths on polyhedra. I, J. Soc. Appl. Ind. Math. 13 (1965), 946-956.
  • [15] L. A. Lyusternik, Convex figures and polyhedra, Dover, New York, 1963.
  • [16] J. W. Moon and L. Moser, Simple paths on polyhedra, Pacific J. Math. 13 (1963), 629-631.
  • [17] T. S. Motzkin, Comonotonecurves and polyhedra, Abstract 111, Bull. Amer. Math. Soc. 63 (1957), 35. 15Here, of course, ''shortest path" is in the sense of 2.6. D. Barnette [24] has produced a simple 3-polytope having only 14 vertices and at the same time having two vertices p and q such that no shortest path from p to q is a W path. He has proved that 14 is the smallest number for which this is possible.
  • [18] E. Steinitz, Polyeder und Raumeinteilungen,Enzyklopadie der Mathematischen Wissenschaften, 3AB 12 (1922), 1-139.
  • [19] E. Steinitz and H. Rademacher, Vorlesungen uber die Theorie der Polyeder, Springer, Berlin, 1934.
  • [20] W. T. Tutte, On Hamiltonian circuits, J. London Math. Soc. 21 (1946), 98-101. 2i# 1A theorem on planar graphs, Trans. Amer. Math. Soc, 82 (1956) 99-116.
  • [22] W. T. Tutte,A non-Hamiltonianplanargraph, Acta Math. Acad. Sci. Hungar. 11 (1960), 371-375.
  • [23] H. Whitney, A theorem on graphs, Ann. of Math. 32 (1931), 378-390.
  • [24] D. Barnette, Wv paths on 3-polytopes,Journal of Combinatorial Theory, to appear.
  • [25] V. Klee and D. Walkup, The d-step conjecture for polytopes of dimension d < 6, to appear.

See also

  • Victor Klee. Paths on polyhedra. I. I [MR 33 #635] J. Soc. Indust. Appl. Math. 13 1965 946--956.