The Euclidean dimension of the join of two cycles

The Euclidean dimension of a graph GG is the smallest integer pp such that the vertices of GG can be represented by points in the Euclidean space RpRp with two points being 1 unit distance apart if and only if they represent adjacent vertices. We show that dim(Cm+Cn)=5dim(Cm+Cn)=5 except that dim(C4+C4)=4dim(C4+C4)=4, dim(C5+C5)=4dim(C5+C5)=4, and dim(C6+C6)=6dim(C6+C6)=6.