2021/10/19 by Collins, Karen, Roberts, Cleo
#05C10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2110.09679
A thrackle is a graph drawing in which every pair of edges meets exactly once. The Thrackle Conjecture (established by John Conway) states that the number of edges of a thrackle cannot exceed the number of its vertices. Cairns, Koussas, and Nikolayevsky (2015) prove that the Thrackle Conjecture holds for great-circle thrackles drawn on the sphere. They also posit that the Thrackle Conjecture can be restated to say that a graph can be drawn as a thrackle drawing in the plane if and only if it admits a great-circle thrackle drawing. We demonstrate that the class of great-circle thrackleable graphs excludes some trees. Thus the informal conjecture from Cairns, Koussas, and Nikolayevsky (2015) is not equivalent to the Thrackle Conjecture.