[Paper Review] Thrackles containing a standard musquash

Grace Misereh, Yuri Nikolayevsky|arXiv (Cornell University)|Jan 21, 2016

Computational Geometry and Mesh Generation4 references3 citations

TL;DR

This paper proves that a thrackle drawing containing a standard musquash (an odd-length cycle drawn with maximal chords on a circle) cannot contain any additional 3- or 5-cycles, except possibly the musquash itself. Using topological and isotopy arguments, the authors show that adding such cycles leads to unavoidable violations of the thrackle condition, where edges would need to cross more than once or fail to maintain proper intersection order.

ABSTRACT

A thrackle is a drawing of a graph in which each pair of edges meets precisely once. Conway's Thrackle Conjecture asserts that a planar thrackle drawing of a graph cannot have more edges than vertices, which is equivalent to saying that no connected component of the graph contains more than one cycle. We prove that a thrackle drawing containing a standard musquash (standard $n$-gonal thrackle) cannot contain any other cycle of length three or five.

Motivation & Objective

• To investigate the structural limitations of thrackle drawings containing a standard musquash, particularly regarding the presence of additional small cycles.
• To determine whether a figure-eight graph formed by a standard musquash and a 3- or 5-cycle can be drawn as a valid thrackle.
• To establish topological obstructions that prevent the existence of such configurations under the thrackle condition.
• To contribute to the broader effort of validating Conway’s Thrackle Conjecture by ruling out specific counterexample candidates.

Proposed method

• Model thrackle drawings on the 2-sphere and analyze them up to isotopy to simplify topological complexity.
• Reduce the problem to the case of a figure-eight graph formed by two cycles sharing a single vertex, focusing on the standard musquash and a potential 3- or 5-cycle.
• Use Reidemeister moves and edge-attachment constraints to enumerate possible configurations of the secondary cycle relative to the musquash.
• Apply the thrackle condition that every pair of edges must cross exactly once, particularly analyzing the crossing order on shared edges.
• Demonstrate that inserting the final edge of a 3- or 5-cycle leads to topological contradictions, such as unreachable vertices or invalid crossing sequences.
• Leverage known results on musquash classification and the non-existence of 4-cycles in thrackles to constrain the analysis.

Experimental results

Research questions

• RQ1 Can a thrackle drawing that contains a standard musquash also contain a 3-cycle not identical to the musquash?
• RQ2 Is it possible for a thrackle to contain both a standard musquash and a 5-cycle, sharing a common vertex?
• RQ3 What topological or combinatorial constraints prevent the existence of such a figure-eight configuration in a valid thrackle?
• RQ4 Does the standard musquash’s symmetry and crossing pattern inherently block the embedding of additional small cycles?
• RQ5 Can the orientation and order of crossings on edges in a thrackled cycle be used to derive obstructions to adding new cycles?

Key findings

• A thrackle drawing containing a standard musquash cannot contain any additional 3-cycle, as such a configuration leads to a topological contradiction in edge crossings.
• Similarly, no additional 5-cycle can coexist with a standard musquash in a valid thrackle drawing due to incompatible crossing order constraints.
• The proof shows that attempting to close a 3- or 5-cycle in the presence of a standard musquash results in a situation where the final edge cannot be drawn without violating the thrackle condition.
• The analysis confirms that the only possible thrackle drawing of a 5-cycle is the standard 5-musquash, up to isotopy.
• The result supports Conway’s Thrackle Conjecture by eliminating a class of potential counterexamples involving figure-eight graphs with a standard musquash and a small cycle.
• The authors establish that even if a 6-cycle and a 3- or 5-cycle can be thrackled separately, their figure-eight union cannot be thrackled, except in specific cases like the theta-graph Θ₃, which is already known to be non-thracklable.