On the circular-l(2, 1)-labelling for strong products of paths and cycles

Let k be a positive integer. A k-circular-L(2, 1)-labelling of a graph G is an assignment f from V(G) to {0, 1, ..., k-1} such that, for any two vertices u and v, |f(u) -f(v)|_k ≥ 2 if u and v are adjacent, and |f(u) -f(v)|_k ≥ 1 if u and v are at distance 2, where |x|_k = min{|x|, k-|x|}. The minimum k such that G admits a k-circular-L(2, 1)-labelling is called the circular-L(2, 1)-labelling number (or just the σ-number) of G, denoted by σ(G). The exact values of σ(P_m ⊗ C_n) and σ(C_m ⊗ C_n) for some m and n have been determined in this study. Finally, it has been concluded that σ(C_m ⊗ C_n) ≤ 13 for n ≥ m ≥ 220.