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Some pictorial remarks on Suzuki’s Brunnian graph. (English) Zbl 0772.57005
Topology ’90, Contrib. Res. Semester Low Dimensional Topol., Columbus/OH (USA) 1990, Ohio State Univ. Math. Res. Inst. Publ. 1, 351-354 (1992).
[For the entire collection see Zbl 0747.00024.]
A Brunnian graph is one for which every proper subgraph can be isotoped into the plane, but the graph itself cannot. Suzuki has given a family \(\theta_ n\) of Brunnian graphs with an algebraic proof that they are not planar. This paper gives an alternative proof, based on converting the 2-vertex graphs \(\theta_ n\) into ‘2-bridge graphs’. A previous paper of the author gives a criterion for whether such a graph is planar, and use of the Burau representation of the braid group is made here to show that the criterion is not satisfied.

57M15 Relations of low-dimensional topology with graph theory
05C10 Planar graphs; geometric and topological aspects of graph theory