Connectivity of Random Cubic Sum Graphs

Consider the set SG(Qk) of all graphs whose vertices are labeled with non-identity elements of the group Qk = Z2 so that there is an edge between vertices with labels a and b if and only if the vertex labeled a + b is also in the graph. Note that edges always appear in triangles, since a + b = c, b + c = a and a + c = b are equivalent statements for Qk. We define the random cubic sum graph SG(Qk, p) to be the probability space over SG(Qk) whose vertex sets are determined by Pr[x ∈ V ] = p with these events mutually independent. As p increases from 0 to 1, the expected structure of SG(Qk, p) undergoes radical changes. We obtain thresholds for some graph properties of SG(Qk, p) as k → ∞. As with the classical random graph, the threshold for connectivity coincides with the disappearance of the last isolated vertex.