Graph coloring with cardinality constraints on the neighborhoods

Authors:
M. -C. Costa;D. De Werra;C. Picouleau;B. Ries
Affiliations:
CEDRIC, ENSTA, Paris, France;EPFL, IMA, Lausanne, Switzerland;CEDRIC, CNAM, Paris, France;EPFL, IMA, Lausanne, Switzerland
Venue:
Discrete Optimization
Year:
2009

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Abstract

Extensions and variations of the basic problem of graph coloring are introduced. The problem consists essentially in finding in a graph G a k-coloring, i.e., a partition V^1,...,V^k of the vertex set of G such that, for some specified neighborhood N@?(v) of each vertex v, the number of vertices in N@?(v)@?V^i is (at most) a given integer h"v^i. The complexity of some variations is discussed according to 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 G is a special tree).