Forwarding and optical indices of 4-regular circulant networks

An all-to-all routing in a graph G is a set of oriented paths of G, with exactly one path for each ordered pair of vertices. The load of an edge under an all-to-all routing R is the number of times it is used (in either direction) by paths of R, and the maximum load of an edge is denoted by π ( G , R ) . The edge-forwarding index π ( G ) is the minimum of π ( G , R ) over all possible all-to-all routings R, and the arc-forwarding index π ? ( G ) is defined similarly by taking direction into consideration, where an arc is an ordered pair of adjacent vertices. Denote by w ( G , R ) the minimum number of colours required to colour the paths of R such that any two paths having an edge in common receive distinct colours. The optical index w ( G ) is defined to be the minimum of w ( G , R ) over all possible R, and the directed optical index w ? ( G ) is defined similarly by requiring that any two paths having an arc in common receive distinct colours. In this paper we obtain lower and upper bounds on these four invariants for 4-regular circulant graphs with connection set { ? 1 , ? s } , 1 < s < n / 2 . We give approximation algorithms with performance ratio a small constant for the corresponding forwarding index and routing and wavelength assignment problems for some families of 4-regular circulant graphs.

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