Facets of the minimum-adjacency vertex coloring polytope

A wireless network employs some portion of the electromagnetic spectrum to establish communications between the transmitter/receiver network antennas, called TRXs. A certain part of the electromagnetic spectrum is licensed to the company operating the network and is divided into discrete channels. Each TRX must operate through one channel, although whenever two TRXs overlap their coverage areas co-channel interference occurs if both are using the same channel, and communications cannot be established within the common area. Moreover, if these conflicting TRXs are assigned to adjacent channels, then the so-called adjacent-channel interference occurs, generating in this case a minor interference only. In a typical scenario, a good channel assignment must avoid co-channel interference and should avoid adjacent-channel interference. Several other constraints arise in practical settings as, e.g., blocked channels and separation constraints (see, e.g., [4, 5, 6, 10]) but in this work we focus on the basic model as stated in the previous paragraph. We are interested in the polyhedral structure generated by such a combinatorial optimization problem, which includes a graph coloring structure with additional considerations on adjacent channels/colors. Based on these observations, we introduce in this work the minimum-adjacency vertex coloring problem and present an initial polyhedral study for it.