A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. A graph G is s-degenerated for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤ s. We prove that an s-degenerated graph G has a total coloring with ∆ + 1 colors if the...

In a two-capacitated spanning tree of a complete graph with a distinguished root vertex v, every component of the induced subgraph on V\{v} has at most two vertices. We give a complete, non-redundant characterization of the polytope defined by the convex hull of the incidence vectors of two-capacitated spanning trees. This polytope is the intersection of the spanning tree polytope on the given ...

Recently, Chen et al. (2009) proposed a finite polyhedral characterization of the convex hull of solutions to general mixed integer linear programs with bounded integer variables and introduced finitely convergent algorithms to solve this class of problems. In this paper, we investigate whether these algorithms provide computational tools that go beyond the convergence results reported in the a...

It is shown how a connected graph and a tree with partially prescribed spectrum can be constructed. These constructions are based on a recent result of Salez that every totally real algebraic integer is an eigenvalue of a tree. This implies that for any (not necessarily connected) graph G, there is a tree T such that the characteristic polynomial P (G, x) of G can divide the characteristic poly...

The λ-backbone coloring is one of the various problems of vertex colorings in graphs. Given an integer λ ≥ 2, a graph G = (V,E), and a spanning subgraph (backbone) H = (V, EH) of G, a λ-backbone coloring of (G,H) is a proper vertex coloring V → {1, 2, ...} of G in which the colors assigned to adjacent vertices in H differ by at least λ. The λ-backbone coloring number BBCλ(G,H) of (G,H) is the s...

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is sdegenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤ s. We prove that an s-degenerate graph G has a total coloring with ∆+1 colors if the maximum degre...

We consider the verification problem of programs containing the following complex features: (1) dynamic creation of parallel threads, (2) synchronisation between parallel threads via global variables, (3) (possibly recursive) procedure calls, and (4) integer variables. The configurations of such programs are represented by terms, and their transitions by term rewriting systems. The novelty of o...

This paper deals with the multiobjective version of the optimal spanning tree problem. More precisely, we are interested in determining the optimal spanning tree according to an Ordered Weighted Average (OWA) of its objective values. We first show that the problem is weakly NP-hard. We then propose different mixed integer programming formulations, according to different subclasses of OWA functi...

We present an application of Integer Programming to the design of arrival routes for aircraft in a Terminal Maneuvering Area (TMA). We generate operationally feasible merge trees of curvatureconstrained routes, using two optimization criteria: (1) total length of the tree, and (2) distance flown along the tree paths. The output routes guarantee that the overall traffic pattern in the TMA can be...

A. We use the group (Z2,+) and two associated homomorphisms, τ0, τ1, to generate all distinct, non-zero pairs of coprime, positive integers which we describe within the context of a binary tree which we denote T . While this idea is related to the Stern-Brocot tree and the map of relatively prime pairs, the parents of an integer pair these trees do not necessarily correspond to the paren...