#### Nonrevisiting paths on surfaces

DiscreteComputGeom
Nonrevisiting Paths on Surfaces 0
H. Pulapaka 0
A. Vince 0
0 Departmentof Mathematics , Universityof Florida, Gainesville,FL 32611 , USA
AbstraeL The nonrevisiting path conjecture for polytopes, which is equivalent to the Hirsch conjecture, is open. However, for surfaces, the nonrevisiting path conjecture is known to be true for polyhedral maps on the sphere, projective plane, toms, and a Klein bottle. Barnette has provided counterexamples on the orientable surface of genus 8 and nonorientable surface of genus 16. In this note the question is settled for all the remaining surface except the connected sum of three copies of the projective plane.
1. Introduction
One of the most well-known open problems in the combinatorial theory of polytopes
is the Hirsch conjecture, which gives an upper bound on the diameter of the graph of a
polytope. The graph o f a polytope P is the 1-skeleton of P. More specifically, the Hirsch
conjecture states that A(d, n) < n - d, where A(d, n) is the maximum diameter among
the graphs of d-dimensional polytopes with n facets. A facet is a (d - 1)-dimensional
face. The Hirsch conjecture was formulated by Hirsch in 1957 and reported by Dantzig
in his book Linear Programming and Extensions [
5
]. The conjecture has implications
for the complexity of linear programming algorithms like the simplex method. Since
the distance between two points on the graph of a polytope P is a lower bound on the
number of iterations of an edge-following algorithm for an LP problem with feasible
region P, the diameter A (d, n) gives the worst-possible complexity for the best-possible
edge-following algorithm. A nice survey on the Hirsch conjecture is the paper by Klee
and Kleindschmidt [
9
].
Equivalent to the Hirsch conjecture is the nonrevisiting path conjecture [
8
] of Klee
and Wolfe. If p is a path in the graph of a polytope, a revisit of p to a face F is a pair of
vertices (x, y) such that p[x, y] N F = {x, y}, where p[x, y] is the path along p from
x to y. In other words, p visits F at x, leaves F, and subsequently, revisits F at y.
Nonrevisiting Path Conjecture. Any two vertices of a polytope P can be joined by a
path that does not revisit any facet of P.
The Nonrevisiting Path Conjecture is known to be true for three-dimensional
polytopes [
1
] and is open in higher dimensions. Klee and Walkup [
10
] showed it to be false,
in general, for unbounded polyhedra. Klee [
7
] has asked about the validity of the
Nonrevisiting Path Conjecture for more general complexes. Since the underlying topological
space of the boundary complex of a polytope is a sphere, it is natural to ask whether
the conjecture is true for cell complexes whose underlying space is a sphere. In this
regard, the conjecture is true for 2-spheres, but there is a counterexample due to Mani
and Walknp [11] for the 3-sphere.
This note concerns the Nonrevisiting Path Conjecture for polyhedral maps. By a
surface S we mean a connected, compact 2-manifold without boundary. These comprise
the orientable surfaces Tg of genus g, which are the connected sums of g toil, and the
nonorientable surfaces Uh, which are the connected sums of h projective planes. Let G
be a graph embedded on a surface S. The closure of a connected component of G \ S is
called aface. If the faces are all simply connected and the intersection of any two distinct
faces is either a common edge, common vertex, or empty, then M = (G, S) is called
a polyhedral map. Two distinct faces that satisfy the condition stated above are said to
meet properly. A surface S has the nonrevisiting path property if, for any polyhedral
map M on S and any two vertices x and y on M, there is a path joining x to y that does
not revisit any face. Recent research has been directed toward the following question.
Question.
Which surfaces possess the nonrevisiting path property?
The nonrevisiting property holds for the sphere [
1
], [
8
], projective plane [
2
], torus [
3
],
and Klein bottle [
6
], [
12
]. However, Barnette [
4
] has recently provided counterexamples
for/'8 and U16. In this note we settle the question for all the remaining surfaces except
U3, the connected sum of three copies of the projective plane. This may also clarify a
misconception [
6
] that the nonrevisiting path property holds for T2, the two-hole torus.
Theorem. The nonrevisiting path property holdsfor the sphere, torus, projective plane,
and Klein bottle; it does not hold for all other surfaces except possibly U3.
The Counterexamples
The proof of the theorem stated in the Introduction requires the construction of
counterexamples for all surfaces except the sphere, projective plane, torus, Klein bottle, and
/-/3. The first counterexample is a polyhedral map M on T2. Figure 1 shows 16 faces. The
polyhedral map M is obtained by identifying like labeled edges of these faces. In order
to conclude that the result is indeed a polyhedral map, it must be verified that:
(1) The neighborhood of each vertex is homeomorphic to a disk (as opposed to, say,
two disks pinched together at that vertex).
NonrevisitingPathson Surfaces 355 C 3
A
F3
A
X
C
O
1
Fq
I
B
B
4
C
O
T
4
v
3
Y
1
2
il
F,
F,
A
D
A
2
(2) Pairs of distinct faces meet properly.
(3) The surface is T2.
It is easy to check that the boundary cycles of the faces have a coherent orientation,
i.e., an orientation such that, for each edge, the directions induced by the two incident
faces are opposite. Thus the surface is orientable. Because the surface has 10 vertices,
28 edges, and 16 faces, the Euler characteristic is g = 10 -- 28 + 16 = --2, which
implies that the genus is (2 - X)/2 = 2. Therefore the surface is T2. Conditions (1) and
(2) above are easily checked since the example is small.
To show that M does not satisfy the nonrevisiting property, we prove that the vertices
labeled x and y in Fig. 1 cannot be joined by a nonrevisiting path. Assume, by way of
contradiction, that p is a nonrevisiting path joining x and y. Because of the symmetry
of M there is no loss of generality in assuming that the vertex adjacent to x along p
is the vertex labeled A. The path p has now left the faces labeled F3 and Fa and has
visited the face F6. Since p is assumed to be nonrevisiting and the vertex y lies on F6,
the remainder of p must also lie on the face F6. There are two ways to get from A to
y along F6, via vertex 2 or via vertex 3. If p passes through vertex 2, then the face F4
is revisited by p; if p passes through vertex 3, then the face F3 is revisited. Either way
leads to a contradiction.
In order to construct a counterexample for the surface Tg, g > 3, let M be a polyhedral
map on the surface Tg_2 such that M has a triangular face F0. The connected sum M # M
of M and M, formed by removing face F0 from M and face F9 of M and identifying
the two___triangularboundary cycles, is a map on the surface Tg..__Moreover,any two faces
of M#M meet properly; otherwise if faces F in M and F ' in M meet improperly, then
either F and F9 meet improperly on M or F ' and F0 meet improperly on M. Both are
impossible because M and M are polyhedral maps. The proof that x and y cannot be
A
A
C
1
v
F
B
3
B
4
B
C
D
4
A
2
D
A
4
j o i n e d by a nonrevisiting path in M # M is identical to the proof for M alone. Thus the
theorem is proved for the orientable surfaces.
For the nonorientable case consider the 17 faces in Fig. 2. As in the orientable case,
identify edges with the same labels to obtain a polyhedral map N. It is easy to verify
that the surface is nonorientable because there is no coherent orientation of the boundary
cycles of the faces. The Euler characteristic is 11 - 30 + 17 = - 2 , so the u n d e r l y i n g
surface is U4, the connected sum of four projective planes (or, equivalently, two Klein
bottles or a torus and Klein bottle). The proof that N does not satisfy the nonrevisiting
path property is identical to the one given for M. Counterexamples for the surfaces
Uh, h > 5, are also obtained in a m a n n e r similar to the orientable case, by taking the
connected sum of N and a polyhedral map N on Uh-4.
1. D.W. Barnette , Wv paths on 3-polytopes , J. Combin. Theory 7 ( 1969 ), 62 - 70 .
2. D. W. Barnette , Wv paths in the projectiveplane , Discrete Math. 62 ( 1986 ), 127 - 131 .
3. D. W. Barnette , Wv Pathson the toms , Discrete Comput. Geom. 5 ( 1990 ), 603 - 608 .
4. D.W. Barnette , A 2-manifoldofgenuseight withoutthe Wv-property , Geom. Dedicata 46 ( 1993 ), 211 - 214 .
5. G. B. Dantzig , Linear Programming and Extensions , PrincetonUniversityPress, Princeton,NJ, 1963 .
6. E. Engelhardt,SomeProblemson Paths in Graphs, Ph.D. Thesis , Universityof Washington,Seattle, WA, 1988 .
7. V. Klee ,Problem 19 , Proc . Colloquium on Convexity, Copenhagen, 1965 .
8. V. Klee , Paths on polyhedraI, SIAM J . 13 ( 1965 ), 946 - 956 ; Paths on polyhedraII, Pacific J. Math. 17 ( 1966 ), 249 - 262 .
9. V. Klee and P. Kleindschmidt , The d-step conjecture and its relatives , Math. Oper. Res . 12 ( 4 ) ( 1987 ), 718 - 755 .
10. V. Klee and D. Walkup , The d-step conjecture for polyhedra of dimension d < 6 , Acta Math . 133 ( 1967 ), 595 - 598 .
I I. P. Mani and D. Waikup , A 3-sphere counterexample to the Wv-path conjecture , Math. Open. Res . 5 ( 4 ) ( 1980 ), 595 - 598 .
12. H. Pulapaka, Nonrevisiting Paths and Cycles on Polyhedral Maps, Ph.D. Thesis , University of Florida, Gainesville, FL, 1995 .