We improve previous lower bounds on the number of simple polygonizations, and other kinds of crossing-free subgraphs, of a set of N points in the plane by analyzing a suitable configuration. We also prove that the number of crossing-free perfect matchings and spanning trees is minimum when the points are in convex position.  2000 Elsevier Science B.V. All rights reserved.

For a graph G and its spanning tree T the backbone chromatic number, BBC(G,T ), is defined as the minimum k such that there exists a coloring c : V (G) → {1, 2, . . . , k} satisfying |c(u) − c(v)| ≥ 1 if uv ∈ E(G) and |c(u)− c(v)| ≥ 2 if uv ∈ E(T ). Broersma et al. [1] asked whether there exists a constant c such that for every triangle-free graphG with an arbitrary spanning tree T the inequali...

For a graph G and its spanning tree T the backbone chromatic number, BBC(G,T ), is defined as the minimum k such that there exists a coloring c : V (G) → {1, 2, . . . , k} satisfying |c(u) − c(v)| ≥ 1 if uv ∈ E(G) and |c(u)− c(v)| ≥ 2 if uv ∈ E(T ). Broersma et al. [1] asked whether there exists a constant c such that for every triangle-free graphG with an arbitrary spanning tree T the inequali...

Given a vertex-weighted connected graphG = (V,E), the maximum weight internal spanning tree (MwIST for short) problem asks for a spanning tree T of G such that the total weight of the internal vertices in T is maximized. The unweighted variant, denoted as MIST, is NPhard and APX-hard, and the currently best approximation algorithm has a proven performance ratio 13/17. The currently best approxi...

We prove the existence of an automorphism-invariant coupling for the wired and the free uniform spanning forests on connected graphs with residually amenable automorphism groups. MSC: 60D05, 05C05, 60B99, 20F32.

We introduce a one-parameter family of massive Laplacian operators (∆)k∈[0,1) defined on isoradial graphs, involving elliptic functions. We prove an explicit formula for minus the inverse of ∆, the massive Green function, which has the remarkable property of only depending on the local geometry of the graph, and compute its asymptotics. We study the corresponding statistical mechanics model of ...

In this paper, we consider the step fixed-charge transportation problem (FCTP) in which a step fixed cost, sometimes called a setup cost, is incurred if another related variable assumes a nonzero value. In order to solve the problem, two metaheuristic, a spanning tree-based genetic algorithm (GA) and a spanning tree-based memetic algorithm (MA), are developed for this NP-hard problem. For compa...