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...

Vertex coloring of a graph is the assignment of labels to the vertices of the graph so that adjacent vertices have different labels. In the case of polyhedral graphs, the chromatic number is 2, 3, or 4. Edge coloring problem and face coloring problem can be converted to vertex coloring problem for appropriate polyhedral graphs. We have been developed an interactive learning system of polyhedra,...

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...

Many classes of graphs where the vertex coloring problem is polynomially solvable are known, the most prominent being the class of perfect graphs. However, the list-coloring problem is NP-complete for many subclasses of perfect graphs. In this work we explore the complexity boundary between vertex coloring and list-coloring on such subclasses of perfect graphs, where the former admits polynomia...

In parallel computing, a valid graph coloring yields a lock-free processing of the colored tasks, data points, etc., without expensive synchronization mechanisms. However, coloring is not free and the overhead can be significant. In particular, for the bipartite-graph partial coloring (BGPC) and distance-2 graph coloring (D2GC) problems, which have various use-cases within the scientific comput...

An L(2, 1)-coloring of a graph G is a coloring of G’s vertices with integers in {0, 1, . . . , k} so that adjacent vertices’ colors differ by at least two and colors of distance-two vertices differ. We refer to an L(2, 1)-coloring as a coloring. The span λ(G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is λ(G), and the hole index ρ(G) of...

If a graph G contains no subgraph isomorphic to some graph H, then G is called H-free. A coloring of a graph G = (V,E) is a mapping c : V → {1, 2, . . .} such that no two adjacent vertices have the same color, i.e., c(u) 6= c(v) if uv ∈ E; if |c(V )| ≤ k then c is a k-coloring. The Coloring problem is to test whether a graph has a coloring with at most k colors for some integer k. The Precolori...

Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of following problems by k: (1) Given a graph G, clique modulator D (a is set vertices, whose removal results in clique) size k for list L(v) colors every v ∈ V(G), decide whether G has proper coloring; (2) pre-coloring λ_P: X → Q ⊆ λ_P can be extended coloring using only from Q. For Problem 1 we des...