This paper presents an extraction and expansion approach for the graph coloring problem. The extraction phase transforms a large graph into a sequence of progressively smaller graphs by removing large independent sets from the graph. The expansion phase starts by generating an approximate coloring for the smallest graph in the sequence. Then it expands the smallest graph by progressively adding...

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 consider worst case time bounds for NP-complete problems including 3-SAT, 3-coloring, 3-edge-coloring, and 3-list-coloring. Our algorithms are based on a common generalization of these problems, called symbol-system satissability or, brieey, SSS 1]. 3-SAT is equivalent to (2,3)-SSS while the other problems above are special cases of (3,2)-SSS; there is also a natural duality transformation f...

A new edge-coloring algorithm for bipartite graphs is presented. This algorithm, based on the framework of the O(m log d + (m/d) log(m/d) log d) algorithm by Makino–Takabatake–Fujishige and the O(m log m) one by Alon, finds an optimal edge-coloring of a bipartite graph with m edges and maximum degree d in O(m log d + (m/d) log(m/d)) time. This algorithm does not require elaborate data structure...

Given fixed 0 = q0 < q1 < q2 < · · · < qk = 1 a constellation in [n] is a scaled translated realization of the qi with all elements in [n], i.e., p, p + q1d, p + q2d, . . . , p + qk−1d, p + d. We consider the problem of minimizing the number of monochromatic constellations in a two coloring of [n]. We show how given a coloring based on a block pattern how to find the number of monochromatic sol...

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...

Multi-coloring problem is a generalization of the well known Graph coloring problem which is known to be NP-hard. Multi-coloring problem can be solved by algorithms designed for the graph coloring problem after transformation of the graphs. However, since the graph transformations increase the size and order of the given graph, in some cases, it may be impractical to solve multi-coloring proble...

We consider a multiagent extension of single-agent graph coloring. Multiple agents hold disjoint autonomous subgraphs of a global graph, and every color used by the agents in coloring the graph has associated cost. In this multiagent graph coloring scenario, we seek a minimum legal coloring of the global graph’s vertices, such that the coloring is also Pareto efficient, socially fair, and indiv...

Ising machines can find optimum or quasi-optimum solutions of combinatorial optimization problems efficiently and effectively. The graph coloring problem, which is one the difficult problems, to assign a color each vertex such that no two vertices connected by an edge have same color. Although methods map problem onto model quadratic unconstrained binary (QUBO) are proposed, none them considers...