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.