We obtain the characterization of saturation class and approximation spaces for perfect spline approximation. Also, a generalized perfect spline approximation is investigated.

Let G be a graph and let c be a coloring of its edges. If the sequence of colors along a walk of G is of the form a1, . . . , an, a1, . . . , an, the walk is called a square walk. We say that the coloring c is squarefree if any open walk is not a square and call the minimum number of colors needed so that G has a square-free coloring a walk Thue number and denote it by πw(G). This concept is a ...

We study the weighted improper coloring problem, a generalization of defective coloring. We present some hardness results and in particular we show that weighted improper coloring is not fixed-parameter tractable when parameterized by pathwidth. We generalize bounds for defective coloring to weighted improper coloring and give a bound for weighted improper coloring in terms of the sum of edge w...

Graph coloring is the de facto standard technique for register allocation within a compiler. In this paper we examine the intuition that a better coloring algorithm results in better register allocation. By replacing the coloring phase of the gcc compiler’s register allocator with an optimal coloring algorithm, we demonstrate both the importance of extending the graph coloring model to better e...

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring, the problem of determining whether a given graph has Grundy number at least k. We also study the variants Weak Grundy Coloring (where the coloring is not necessarily proper) and Connected Grundy Col...

We study the weighted improper coloring problem, a generalization of defective coloring. We present some hardness results and in particular we show that weighted improper coloring is not fixed-parameter tractable when parameterized by pathwidth. We generalize bounds for defective coloring to weighted improper coloring and give a bound for weighted improper coloring in terms of the sum of edge w...