In many applications of graph colouring the sizes of colour classes should not be too large. For example, in scheduling jobs (some of which could be performed at the same time), it is not good if the resulting schedule requires many jobs to occur at some specific time. An application of this type is discussed in [8]. A possible formalization of this restriction is the notion of equitable colour...

This paper describes an exact algorithm for the Equitable Coloring Problem, based on the well known DSatur algorithm for the classic Coloring Problem with new pruning rules specifically derived from the equity constraint. Computational experiences show that our algorithm is competitive with those known in literature.

The first problem we deal with in this dissertation concerns long induced cycles in the d-dimensional cube, i.e. the graph on d-tuples of binary digits with pairs of vertices differing in exactly one coordinate as edges. Such cycles were introduced by Kautz as a type of error-checking codes (called the snake-in-the-box codes) for a certain analogue-to-digital conversion system. We give a simple...

Let G be an intuitionistic fuzzy graph. Let u and v be two vertices of G. A subset D of V is called a fuzzy equitable dominating set if every v ∈ V − Dthere exist a vertex u ∈ D such that uv ∈ E(G) and |deg(u) − deg(v)| = 1 where deg(u) denotes the degree of vertex u and deg(v) denotes the degree of vertex v and μ2(vi, vj ) = μ1(vi) ∧ μ1(vj ), γ2(vi, vj ) = γ1(vi) ∨ γ1(vj ). The minimum cardina...

An employee transporting problem is described and a set partitioning model is developed. An investigation of the model leads to a knapsack problem as a surrogate problem. Finding a partition corresponding to the knapsack problem provides a solution to the problem. An exact algorithm is proposed to obtain a partition (subset-vehicle combination) corresponding to the knapsack solution. It require...

Chen, Lih, and Wu conjectured that for r≥3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are Kr,r (for odd r) and Kr+1. If true, this would be a strengthening of the Hajnal–Szemerédi Theorem and Brooks’ Theorem. We extend their conjecture to disconnected graphs. For r≥6 the conjecture says the following: If an r-colorable graph G with maximum degree...

The official handicapping system of the Royal Canadian Golf Association (RCGA) is very similar to the handicapping system of the United States Golf Association (USGA). Although these handicapping systems are complex and have been carefully studied, the systems do not take statistical theory into account. In 2000, the Handicap Research Committee of the RCGA was formed and challenged with the tas...

An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most 1. A d-degenerate graph is a graph G in which every subgraph has a vertex with degree at most d. A star Sm with m rays is an example of a 1-degenerate graph with maximum degree m that needs at least 1 + m/2 colors for an equitable coloring. Our main result is that every n-...

Location problems can be considered as multiple criteria models where for each client (spatial unit) there is de®ned an individual objective function, which measures the e€ect of a location pattern with respect to the client satisfaction (e.g., it expresses the distance or travel time between the client and the assigned facility). This results in a multiple criteria model taking into account th...