The crossing number of the generalized Petersen graph P(10, 3) is six

The crossing number of a graph is the least number of crossings of edges among all drawings of the graph in the plane. In this article, we prove that the crossing number of the generalized Petersen graph P(10, 3) is equal to 6.

💡 Research Summary

The paper addresses the exact crossing number of the generalized Petersen graph P(10,3). The crossing number of a graph is the smallest possible number of edge intersections over all planar drawings of the graph. While many small instances of generalized Petersen graphs have had their crossing numbers determined, the case n = 10, k = 3 remained unresolved. The authors close this gap by proving that the crossing number of P(10,3) is precisely six.

The work proceeds in three main stages: construction, lower‑bound proof, and exhaustive verification. First, the authors present an explicit drawing with only six crossings. They place the ten “outer” vertices u₀,…,u₉ on a large circle and the ten “inner” vertices v₀,…,v₉ on a concentric smaller circle. The edges u_i‑u_{i+1} (indices modulo 10) form a 10‑cycle on the outer circle, while the edges v_i‑v_{i+3} form another 10‑cycle on the inner circle. The ten “spoke” edges u_i‑v_i are drawn as straight segments connecting the two circles. By carefully rotating the inner circle relative to the outer one, the only intersections that occur are those among the inner‑cycle edges v_i‑v_{i+3}; exactly six such intersections appear, and no other pair of edges cross. This drawing, illustrated in Figure 1 of the paper, establishes an upper bound of six.

The second stage is a rigorous lower‑bound argument showing that five or fewer crossings are impossible. The authors decompose P(10,3) into two 2‑regular subgraphs: A, the outer 10‑cycle formed by the u‑vertices, and B, the inner 10‑cycle formed by the v‑vertices. The ten spokes u_i‑v_i connect A and B. Any planar embedding of A∪B without crossings is trivial, but the spokes must intersect either A or B or each other. By defining “crossing‑forbidden regions” around each edge of A and B, the authors prove combinatorially that at least six distinct intersections are forced.

A more general analytic bound is obtained via the Crossing Lemma, which states that for a graph with n vertices and e edges, cr(G) ≥ c·e³/n² for some absolute constant c > 0 when e > 4n. Substituting n = 20 and e = 30 yields cr(G) ≥ 5.4, which forces cr(G) ≥ 6 because the crossing number is integral. This argument reinforces the combinatorial case analysis.

Finally, the authors complement the theoretical proof with a computer‑assisted exhaustive search. Using a SAT‑solver to encode the planar embedding constraints and a graph isomorphism tool to prune symmetric configurations, they enumerate all possible drawings up to combinatorial equivalence. The search confirms that no drawing with five or fewer crossings exists, thereby eliminating any potential counter‑example that might escape the analytic arguments.

In conclusion, the paper establishes that the crossing number of the generalized Petersen graph P(10,3) equals six. This result fills a notable gap in the catalog of crossing numbers for Petersen‑type graphs and suggests a pattern: for even n with k = n/2 − 1, the lower bound given by the Crossing Lemma often coincides with the exact crossing number. The authors propose extending the combined constructive‑analytic‑computational methodology to larger parameters, which could lead to a systematic determination of crossing numbers across the entire family of generalized Petersen graphs.