[Paper Review] The conjecture cr(C_m imes C_n)=(m-2)n is true for all but finitely many n, for each m

Lev Glebsky, Gelasio Salazar|ArXiv.org|Sep 26, 2000

Advanced Combinatorial Mathematics5 references4 citations

TL;DR

This paper proves that the Harary-Kainen-Schwenk (HKS) conjecture — that the crossing number of the Cartesian product $C_m \times C_n$ is exactly $(m-2)n$ — holds for all but finitely many $n$ for each fixed $m \geq 3$. Using a novel adaptation of the theory of arrangements and topological arguments based on the Jordan Curve Theorem, the authors establish that for $n \geq \frac{m}{2}\left(\frac{(m+3)^2}{2} + 1\right)$, the crossing number is precisely $(m-2)n$, resolving the conjecture asymptotically for each $m$. The proof hinges on associating at least $m-2$ crossings to each of the $n$ red $m$-cycles in a robust drawing, ensuring no crossing is double-counted.

ABSTRACT

It has been long congectured that the crossing number of $C_m imes C_n$ is $(m-2)n$ for $2=(m/2)((m+3)^2/2+1)$.The proof is largely based on the theory of arrangements introduced by Adamsson and further developed by Adamsson and Richter.

Motivation & Objective

• To resolve the long-standing Harary-Kainen-Schwenk (HKS) conjecture that the crossing number of $C_m \times C_n$ is $(m-2)n$ for all $m,n$ with $n \geq m \geq 3$.
• To establish that the HKS conjecture holds for all but finitely many $n$, for each fixed $m \geq 3$, by proving a tight lower bound on the crossing number.
• To develop and apply a topological-combinatorial framework based on robust drawings and circular arrangements to derive this bound.
• To extend the result to a broader class of 4-regular graphs, called $(m,n)$-graphs, showing the same crossing number lower bound holds.

Proposed method

• The proof uses induction on $n$, with the base case $n = n_0 = \frac{(m+3)^2}{2} + 1$, relying on the fact that $\text{cr}(C_m \times C_n) \geq \min\{(m-2)n, m(n - n_0)\}$.
• A key technique is the definition of a 'robust drawing' — a drawing satisfying topological conditions ensuring that $m$-cycles interact in a way that prevents degeneracy and enables consistent crossing counting.
• The authors associate at least $m-2$ crossings to each of the $n$ red $m$-cycles in a robust drawing, ensuring no crossing is counted more than once.
• Topological tools, particularly the Jordan Curve Theorem and analysis of cyclic orderings of curves and arcs, are used to prove that such associations are possible and disjoint.
• The proof leverages the theory of $(m,n)$-circular arrangements, originally developed by Adamsson and Richter, though the authors do not use the term explicitly.
• An improved version of the main result is derived by combining the inductive argument with the known general lower bound $\text{cr}(C_m \times C_n) \geq \frac{1}{2}(m-2)n$.

Experimental results

Research questions

• RQ1 Does the HKS conjecture $\text{cr}(C_m \times C_n) = (m-2)n$ hold for all but finitely many $n$, for each fixed $m \geq 3$?
• RQ2 Can the crossing number of $C_m \times C_n$ be bounded below by $(m-2)n$ for sufficiently large $n$, using topological and combinatorial arguments?
• RQ3 What structural properties of drawings (specifically 'robustness') ensure that crossings can be uniquely associated to $m$-cycles without overcounting?
• RQ4 How does the theory of circular arrangements apply to proving crossing number bounds in $C_m \times C_n$?
• RQ5 Can the bound be improved by incorporating known general lower bounds on crossing numbers?

Key findings

• The HKS conjecture holds for all $n \geq \frac{m}{2}\left(\frac{(m+3)^2}{2} + 1\right)$ and $m \geq 3$, with $\text{cr}(C_m \times C_n) = (m-2)n$.
• The proof establishes that every robust drawing of $C_m \times C_n$ has at least $(m-2)n$ crossings, which implies the main result via induction.
• The authors show that in a robust drawing, at least $m-2$ crossings can be uniquely associated with each of the $n$ red $m$-cycles, and no crossing is shared between cycles.
• An improved version of the main theorem is derived, showing $\text{cr}(C_m \times C_n) = (m-2)n$ holds for $n \geq \left(\frac{m}{4} + \frac{1}{2}\right)\left(\frac{(m+3)^2}{2} + 1\right)$, using the general lower bound $\text{cr}(C_m \times C_n) \geq \frac{1}{2}(m-2)n$.
• The result extends to a broader class of 4-regular graphs called $(m,n)$-graphs, where $\text{cr}(G) \geq (m-2)n$ for $n \geq \frac{m}{2}\left(\frac{(m+3)^2}{2} + 1\right)$.
• The paper leaves open the case for $n < \frac{m}{2}\left(\frac{(m+3)^2}{2} + 1\right)$ when $m \geq 8$, though known bounds remain below the conjectured value.