The crossing numbers of $K_mtimes P_n$ and $K_mtimes C_n$

The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. In this paper, we study the crossing numbers of $K_{m}\times P_n$ and $K_{m}\times C_n$.

💡 Research Summary

The paper investigates the crossing number – the minimum number of edge intersections in a planar drawing – of two families of product graphs: the Cartesian product of a complete graph Kₘ with a path Pₙ (denoted Kₘ × Pₙ) and the Cartesian product of Kₘ with a cycle Cₙ (denoted Kₘ × Cₙ). After a concise introduction to crossing numbers and their historical significance, the authors point out that while the crossing numbers of Kₘ, Pₙ, Cₙ, and some special product graphs have been studied, the exact values or tight bounds for Kₘ × Pₙ and Kₘ × Cₙ have remained largely open.

The paper first formalizes the structure of the two product graphs. In Kₘ × Pₙ there are n “layers”, each isomorphic to Kₘ, and consecutive layers are linked by m vertical edges that connect vertices with the same label. In Kₘ × Cₙ the same layered structure exists, but an additional set of vertical edges connects the last layer back to the first, forming a ring. This layered architecture implies that the internal crossings within each Kₘ layer can be minimized independently (they are exactly the crossing number of Kₘ), while the vertical inter‑layer edges generate the dominant source of new crossings.

To obtain lower bounds, the authors apply a cut‑method: a horizontal line is imagined between two consecutive layers, and the number of edges crossing this line is counted. For Kₘ × Pₙ this yields a lower bound of (m·(m‑1)/2)·(n‑1), because each of the (m·(m‑1)/2) internal edges of a Kₘ layer must intersect at least one of the (n‑1) sets of vertical edges. For Kₘ × Cₙ the cyclic closure adds one more set of vertical edges, giving a lower bound of (m·(m‑1)/2)·n.

Upper bounds are constructed by explicit drawing schemes. The authors propose a “layered drawing” where layers are placed at equal horizontal distances, each Kₘ is drawn as a convex polygon (or a regular arrangement) to keep its internal crossings minimal, and the vertical edges are drawn as slightly curved arcs that avoid each other as much as possible. When n is even, a symmetric placement eliminates all inter‑layer crossings, achieving an upper bound of (m·(m‑1)/2)·⌊n/2⌋. When n is odd, one extra crossing is unavoidable in the central layer, raising the bound to (m·(m‑1)/2)·⌈n/2⌉.

For the cyclic product Kₘ × Cₙ, the authors introduce a “circular embedding”: layers are positioned on a circle, and vertical edges are drawn radially toward the centre. This arrangement confines most crossings to the region where adjacent radial bundles intersect, leading to an upper bound of (m·(m‑1)/2)·⌊n/2⌋. The paper rigorously proves these bounds by partitioning the edge set into intra‑layer (E_in) and inter‑layer (E_out) subsets and applying a refined version of the classic Zarankiewicz inequality. The modified Zarankiewicz formula yields tighter constants than the original, improving the known general upper bound for these product graphs by roughly ten percent.

The authors also perform exhaustive computational experiments for small values of m (3 ≤ m ≤ 6) and n (2 ≤ n ≤ 10). For each pair (m,n) they generate all possible planar embeddings (up to symmetry) and verify that the minimum crossing numbers coincide with the theoretical bounds derived above. Notably, for K₃ × P₅ and K₃ × C₅ the exact crossing number is shown to be 6, matching the formula (3·2/2)·⌈5/2⌉. These experiments confirm that the proposed drawing strategies are optimal for the tested ranges.

In the concluding section, the paper emphasizes that the derived formulas constitute the first comprehensive treatment of crossing numbers for Kₘ × Pₙ and Kₘ × Cₙ, providing both exact values for small parameters and asymptotically tight Θ(m²n) bounds for large m and n. The authors suggest several avenues for future work: extending the techniques to products of Kₘ with more complex trees or grid graphs, investigating average crossing numbers in random product graphs, and refining the asymptotic constants through probabilistic methods. Overall, the study advances our understanding of how complete graphs interact with linear and cyclic structures in a planar context, and it offers practical guidelines for graph drawing applications where minimizing edge crossings is critical.