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)π(G,R). The edge-forwarding index π(G)π(G) is the minimum of π(G,R)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)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)w(G) is defined to be the minimum of w(G,R)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}, 1