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Wireless cellular networks are often modelled as different regular grids and the channel assignment problem for interference avoidance is formulated as a coloring problem of the grid graph, where channels assigned to interfering stations at distance i must be at least δi apart, while the same channel can be reused in stations whose distance is at least σ. In this paper, we consider the channel assignment problem in a class of wireless cellular networks modelled as such regular grids which can be obtained by adding two edges to connect two pair of diagonal vertices of every square cell in the square grids. We confine our discussion to the case that the co-channel reuse distance σ is 4. We present one channel assignment algorithm for the case where the minimum channel separation δi is 1 for all but adjacent stations and δ1 ≥ 3 for adjacent stations, which is proved to be optimal. Also we present one potential optimal algorithm for the case that the minimum channel separation δi is σ-i.