Jan van den Heuvel

Finding paths between 3-colorings

Given a 3-colorable graph G together with two proper vertex 3-colorings α and β of G, consider the following question: is it possible to transform α into β by recoloring vertices of G one at a time, making sure that all intermediate colorings are proper 3-colorings? We prove that this question is answerable in polynomial time. We do so by characterizing the instances G, α, β where the transformation is possible; the proof of this characterization is via an algorithm that either finds a sequence of recolorings between α and β, or exhibits a structure which proves that no such sequence exists. In the case that a sequence of recolorings does exist, the algorithm uses O(|V(G)|2) recoloring steps and in many cases returns a shortest sequence of recolorings. We also exhibit a class of instances G, α, β that require Ω(|V(G)|2) recoloring steps.

List Colouring Squares of Planar Graphs

Abstract In 1977, Wegner conjectured that the chromatic number of the square of every planar graph G with maximum degree Δ ⩾ 8 is at most ⌈ 3 2 Δ ⌉ + 1 . We show that it is at most 3 2 Δ ( 1 + o ( 1 ) ) , and indeed this is true for the list chromatic number.

A Linear Bound On The Diameter Of The Transportation Polytope

We prove that the combinatorial diameter of the skeleton of the polytope of feasible solutions of any m×n transportation problem is at most 8(m+n−2).

Heavy cycles in weighted graphs

An (edge-)weighted graph is a graph in which each edge e is assigned a nonnegative real number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges, and an optimal cycle is one of maximum weight. The weighted degree w(v) of a vertex v is the sum of the weights of the edges incident with v. The following weighted analogue (and generalization) of a well-known result by Dirac for unweighted graphs is due to Bondy and Fan. Let G be a 2-connected weighted graph such that w(v) ≥ r for every vertex v of G. Then either G contains a cycle of weight at least 2r or every optimal cycle of G is a Hamilton cycle. We prove the following weighted analogue of a generalization of Diracs result that was first proved by Posa. Let G be a 2-connected weighted graph such that w(u)+w(v) ≥ s for every pair of nonadjacent vertices u and v. Then G contains either a cycle of weight at least s or a Hamilton cycle. Examples show that the second conclusion cannot be replaced by the stronger second conclusion from the result of Bondy and Fan. However, we characterize a natural class of edge-weightings for which these two conclusions are equivalent, and show that such edge-weightings can be recognized in time linear in the number of edges.

Algorithmic Aspects of a Chip-Firing Game

Algorithmic aspects of a chip-firing game on a graph introduced by Biggs are studied. This variant of the chip-firing game, called the dollar game, has the properties that every starting configuration leads to a so-called critical configuration. The set of critical configurations has many interesting properties. In this paper it is proved that the number of steps needed to reach a critical configuration is polynomial in the number of edges of the graph and the number of chips in the starting configuration, but not necessarily in the size of the input. An alternative algorithm is also described and analysed.

Fire containment in planar graphs

In a graph G, a fire starts at some vertex. At every time step, firefighters can protect up to k vertices, and then the fire spreads to all unprotected neighbors. The k-surviving rate of G is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. For a given class of graphs , we are interested in the minimum value k such that for some constant and all , (i.e., such that linearly many vertices are expected to be saved in every graph from ). In this note, we prove that for planar graphs this minimum value is at most 4, and that it is precisely 2 for triangle-free planar graphs.

Transversals of subtree hypergraphs and the source location problem in digraphs

This invention relates to novel hydrophilic, crosslinked, polyurea-urethane, solid, discrete particles formed by adding in droplet form a hydrophilic prepolymer comprising an isocyanate-capped polyol or mixtures thereof wherein said polyol or mixture of polyols has a reaction functionality greater than two, the total of said polyol present having an ethylene oxide content of at least 40 weight percent, before capping to a large excess of a stirred aqueous reactant. The resultant solid particles can be molded into various shapes as desired or used for coating or adhesive applications.

Using laplacian eigenvalues and eigenvectors in the analysis of frequency assignment problems

A Frequency Assignment Problem (FAP) is the problem that arises when frequencies have to be assigned to a given set of transmitters so that spectrum is used efficiently and the interference between the transmitters is minimal. In this paper we see the frequency assignment problem as a generalised graph colouring problem, where transmitters are presented by vertices and interaction between two transmitters by a weighted edge. We generalise some properties of Laplacian matrices that hold for simple graphs. We investigate the use of Laplacian eigenvalues and eigenvectors as tools in the analysis of properties of a FAP and its generalised chromatic number (the so-called span).

Best Monotone Degree Conditions for Graph Properties: A Survey

We survey sufficient degree conditions, for a variety of graph properties, that are best possible in the same sense that Chvátal’s well-known degree condition for hamiltonicity is best possible.

Finding paths between graph colourings : computational complexity and possible distances.

Abstract Suppose we are given a graph G together with two proper vertex k-colourings of G, α and β. How easily can we decide whether it is possible to transform α into β by recolouring vertices of G one at a time, making sure we always have a proper k-colouring of G? We prove a dichotomy theorem for the computational complexity of this decision problem: for values of k ⩽ 3 the problem is polynomial-time solvable, while for any fixed k ⩾ 4 it is PSPACE-complete. What is more, we establish a connection between the complexity of the problem and its underlying structure: we prove that for k ⩽ 3 the minimum number of necessary recolourings is polynomial in the size of the graph, while for k ⩾ 4 instances exist where this number is superpolynomial.