## Graph coloring with cardinality constraints on the neighborhoods

### Costa, Marie-Christine ; de Werra, Dominique ; Picouleau, Christophe ; Ries, Bernard

### In: discrete Optimization, 2009, vol. 6, no. 4, p. 362-369

Extensions and variations of the basic problem of graph coloring are introduced. The problem consists essentially in finding in a graph a k-coloring, i.e., a partition (V_1,\cdots,V_k) of the vertex set of G such that, for some specified neighborhood \tilde|{N}(v) of each vertex v, the number of vertices in \tilde|{N}(v)\cap V_i is (at most) a given integer h_i^v. The complexity of some... More

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- Extensions and variations of the basic problem of graph coloring are introduced. The problem consists essentially in finding in a graph a k-coloring, i.e., a partition (V_1,\cdots,V_k) of the vertex set of G such that, for some specified neighborhood \tilde|{N}(v) of each vertex v, the number of vertices in \tilde|{N}(v)\cap V_i is (at most) a given integer h_i^v. The complexity of some variations is discussed according to \tilde|{N}(v), which may be the usual neighbors, or the vertices at distance at most 2, or the closed neighborhood of v (v and its neighbors). Polynomially solvable cases are exhibited (in particular when is a special tree).