On a class of Hamiltonian polytopes (original) (raw)

Hamiltonian circuits in prisms over certain simple 3-polytopes

Discrete Mathematics, 1978

In this paper it is shown that the prisr~ eve: cyclically 4-connect simple 3-polytopes ndmit Hamiltonian circuits. It is also shown that if P is a simple 3-polytopc all of whose faces are polygons with six sides or less than the prism over P admits a Hamiltonian circuit.

On Pedigree Polytopes and Hamiltonian Cycles

Electronic Notes in Discrete Mathematics, 2003

In this paper we define a combinatorial object called a pedigree, and study the corresponding polytope, called the pedigree polytope. Pedigrees are in one-to-one correspondence with the Hamiltonian cycles on K n . Interestingly, the pedigree polytope seems to differ from the standard tour polytope, Q n with respect to the complexity of testing whether two given vertices of the polytope are nonadjacent. A polynomial time algorithm is given for nonadjacency testing in the pedigree polytope, whereas the corresponding problem is known to be NP-complete for Q n . We also discuss some properties of the pedigree polytope and illustrate with examples.

Are all simple 4-polytopes Hamiltonian?

Israel Journal of Mathematics, 1983

We construct an extensive family of non-Hamiltonian, 4-regular, 4-connected graphs and show that none of these graphs is the graph of a simple 4-polytope.

The hamiltonian circuit polytope

2013

The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems. We study the polytope by establishing its dimension, developing tools for the identification of facets, and using these tools to derive several families of facets. The tools include necessary and sufficient conditions for an inequality to be facet defining, and an algorithm for generating all undominated circuits. We use a novel approach to identifying families of facetdefining inequalities, based on the structure of variable indices rather than on subgraphs such as combs or subtours. This leads to our main result, a hierarchy of families of facet-defining inequalities and polynomial-time separation algorithms for them. *

Partial monotonizations of Hamiltonian cycle polytopes: dimensions and diameters

Discrete Applied Mathematics, 2000

In this paper we study partial monotonizations and level polytopes of the Hamiltonian Cycle Polytope, also called the symmetric Traveling Salesman Polytope. The kth Level Polytope is the convex hull of the characteristic vectors corresponding to sets of k edges in Kn that can be extended to Hamiltonian cycles (n¿3). For 06 6k, the -monotonization of the kth Level Polytope is the convex hull of the characteristic vectors corresponding to sets of at least and at most k edges in Kn that can be extended to Hamiltonian cycles (n¿3). It is shown that for ¡ k; -monotonizations of level polytopes are full dimensional. We give upper and lower bounds for the diameters of the -monotonizations and determine the number of 0-faces of the level polytopes and -monotonizations. The main result of this paper is a proof that the diameter of the monotone Hamiltonian Cycle Polytope and the monotone Hamiltonian Path Polytope are each Â(log n). ?

Hamiltonicity in (0–1)-polyhedra

Journal of Combinatorial Theory, Series B, 1984

We show that if P& IR" is a polyhedron, all of whose vertices have (0-1)valued coordinates, then (i) if G(P) is bipartite, then G(P) is a hypercube; (ii) if G(P) is nonbipartite, then G(P) is hamihon connected. It is shown that if P&I?" has (0-1)valued vertices and is of dimension d (<n) then there exists a polyhedron P' g IRd having (0-1)valued vertices such that G(P) 1 G(P'). Some combinatorial consequences of these results are also discussed.

Hamiltonicity and combinatorial polyhedra

Journal of Combinatorial Theory, Series B, 1981

We say that a polyhedron with O-1 valued vertices is combinatorial if the midpoint of the line joining any pair of nonadjacent vertices is the midpoint of the line joining another pair of vertices. We show that the class of combinatorial polyhedra includes such well-known classes of polyhedra as matching polyhedra, matroid basis polyhedra, node packing or stable set polyhedra and permutation polyhedra. We show the graph of a combinatorial polyhedron is always either a hypercube (i.e., isomorphic to the convex hull of a k-dimension unit cube) or else is hamilton connected (every pair of nodes is the set of terminal nodes of a hamilton path). This imfilies several earlier results concerning special cases of combinatorial polyhedra.

On the simplicial 3-polytopes with only two types of edges

Discrete Mathematics, 1984

For some families of graphs of simplicial 3-polytopes with two types of edges structural properties are described, for other ones their cardinality is determined. Griinbaum and Motzkin [3], Griinbaum and Zaks [4], and Malkevitch [6] investigated the structural properties of trivalent planar graphs with at most two types of faces. It seems that the knowledge of the structure of such graphs is useful for other reasons as well (cf., e.g. Griinbaum [1, 2], Jucovi~ [5], Owens [7], Zaks [8]). The dual problem may be formulated as follows: Characterize simplicial planar graphs with at most two types of vertices. (A planar graph is simplicial if all its faces are triangles.)