Exact defective colorings of graphs
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An exact (k,d)-coloring of a graph G is a coloring of its vertices with k colors such that each vertex v is adjacent to exactly d vertices having the same color as v. The exact d-defective chromatic number, denoted χ_d^=(G), is the minimum k such that there exists an exact (k,d)-coloring of G. In an exact (k,d)-coloring, which for d=0 corresponds to a proper coloring, each color class induces a d-regular subgraph. We give basic properties for the parameter and determine its exact value for cycles, trees, and complete graphs. In addition, we establish bounds on χ_d^=(G) for all relevant values of d when G is planar, chordal, or has bounded treewidth. We also give polynomial-time algorithms for finding certain types of exact (k,d)-colorings in cactus graphs and block graphs. Our main result is on the computational complexity of d-EXACT DEFECTIVE k-COLORING in which we are given a graph G and asked to decide whether χ_d^=(G) ≤ k. Specifically, we prove that the problem is NP-complete for all d ≥ 1 and k ≥ 2.
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