We study the problem of fixturing a chain of hinged objects in a given placement with frictionless point contacts. We define the notions of immobility and robust immobility—which are comparable to second and first order immobility for a single object—to capture the intuitive requirement for a fixture of a chain of hinged objects; robust immobility differs from immobility in that it additionally...

We give two combinatorial proofs of a product formula for the number of spanning trees of the n-dimensional hypercube. The first proof is based on the assertion that if one chooses a uniformly random rooted spanning tree of the hypercube and orient each edge from parent to child, then the parallel edges of the hypercube get orientations which are independent of one another. This independence pr...

We study path auction games in which multiple edges may be owned by the same agent in this paper. The edge costs and the set of edges owned by the same agent are privately known to the owner of the edges. In this setting, we show that, assuming that losing agents always have 0 payoff, there is no individual rational, strategyproof mechanism in which only edge costs are reported. If the agents a...

A c-edge-colored multigraph has each edge colored with one of the c available colors and no two parallel edges have the same color. A proper hamiltonian path is a path containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper hamiltonian path, depending on several parameter...

An integral sequence d = (d1, d2, . . . , dn) is hypergraphic if there is a simple hypergraph H with degree sequence d, and such a hypergraph H is a realization of d. A sequence d is r-uniform hypergraphic if there is a simple r-uniform hypergraph with degree sequence d. Similarly, a sequence d is r-uniformmulti-hypergraphic if there is an r-uniformhypergraph (possibly with multiple edges) with...

A k-planar graph is a graph that can be drawn in the plane such that every edge is crossed at most k times. For k ≤ 4, Pach and Tóth [19] proved a bound of (k + 3)(n− 2) on the total number of edges of a k-planar graph, which is tight for k = 1, 2. For k = 3, the bound of 6n− 12 has been improved to 11 2 n− 11 in [18] and has been shown to be optimal up to an additive constant for simple graphs...

A cubic graph H is called a Kotzig graph if H has a circuit double cover consisting of three Hamilton circuits. It was first proved by Goddyn that if a cubic graph G contains a spanning subgraph H which is a subdivision of a Kotzig graph then G has a circuit double cover. A spanning subgraph H of a cubic graph G is called a Kotzig frame if the contracted graph G/H is even and every non-circuit ...

There has been recent interest in combinatorial versions of classical theorems in topology. In particular, Stahl [S] and Little [3] have proved discrete versions of the Jordan Curve Theorem. The classical theorem states that a simple closed curve y separates the 2-sphere into two connected components of which y is their common boundary. The statements and proofs of the combinatorial versions in...