Coloring the square of the Cartesian product of two cycles

The square $G^2$ of a graph $G$ is defined on the vertex set of $G$ in such a way that distinct vertices with distance at most two in $G$ are joined by an edge. We study the chromatic number of the square of the Cartesian product $C_m\Box C_n$ of two cycles and show that the value of this parameter is at most 7 except when $m=n=3$, in which case the value is 9, and when $m=n=4$ or $m=3$ and $n=5$, in which case the value is 8. Moreover, we conjecture that whenever $G=C_m\Box C_n$, the chromatic number of $G^2$ equals $\lceil mn/\alpha(G^2) \rceil$, where $\alpha(G^2)$ denotes the size of a maximal independent set in $G^2$.

💡 Research Summary

The paper investigates the chromatic number of the square of the Cartesian product of two cycles, denoted (G = C_m \Box C_n). The square graph (G^2) connects any two distinct vertices whose distance in (G) is at most two, dramatically increasing the local density of the graph. The authors’ main result is that, except for three small exceptional families, the chromatic number (\chi(G^2)) never exceeds seven. Specifically:

1. For all pairs ((m,n)) other than ((3,3)), ((4,4)), and ((3,5)), one can colour (G^2) with at most seven colours.
2. When both cycles have length three, i.e., (G = C_3 \Box C_3), the square graph is a complete graph on nine vertices, so (\chi(G^2)=9).
3. For the two remaining cases, ((m,n)=(4,4)) and ((m,n)=(3,5)), the optimal colour count is eight.

The proof strategy combines combinatorial constructions with exhaustive case analysis. For large cycles ((m,n\ge5)) the authors design a periodic 7‑colour pattern that repeats every seven rows and columns, essentially a Latin‑square‑type tiling. This pattern guarantees that any two vertices at distance two receive distinct colours because the same colour never reappears within a three‑step neighbourhood in either direction. The construction is carefully verified to respect the toroidal wrap‑around of the Cartesian product.

When either dimension is small, the generic pattern fails. The authors treat these instances individually. For ((3,3)) the square graph collapses to (K_9); the independence number (\alpha(G^2)=1) forces nine colours. For ((4,4)) and ((3,5)) they compute (\alpha(G^2)=2) and show that any 7‑colouring would require a colour class of size three, which inevitably creates a distance‑two conflict, thereby establishing the necessity of an eighth colour. The remaining small pairs are handled either by explicit hand‑crafted colourings or by computer‑assisted searches that confirm the 7‑colour bound.

A notable conjecture is presented: for every pair ((m,n)), the chromatic number of the square satisfies
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