Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
TY - JOUR
T1 - The nonsolvability by radicals of generic 3-connected planar Laman graphs.
AU - Power, Stephen C.
AU - Owen, J. C.
N1 - Copyright 2006, American Mathematical Society RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics
PY - 2007
Y1 - 2007
N2 - We show that planar embeddable -connected Laman graphs are generically non-soluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity. Formally, a Laman graph is a maximally independent graph, that is, one that satisfies the vertex-edge count together with a corresponding inequality for each subgraph. The following main theorem of the paper resolves a conjecture of Owen (1991) in the planar case. Let be a maximally independent -connected planar graph, with more than 3 vertices, together with a realisable assignment of generic distances for the edges which includes a normalised unit length (base) edge. Then, for any solution configuration for these distances on a plane, with the base edge vertices placed at rational points, not all coordinates of the vertices lie in a radical extension of the distance field.
AB - We show that planar embeddable -connected Laman graphs are generically non-soluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity. Formally, a Laman graph is a maximally independent graph, that is, one that satisfies the vertex-edge count together with a corresponding inequality for each subgraph. The following main theorem of the paper resolves a conjecture of Owen (1991) in the planar case. Let be a maximally independent -connected planar graph, with more than 3 vertices, together with a realisable assignment of generic distances for the edges which includes a normalised unit length (base) edge. Then, for any solution configuration for these distances on a plane, with the base edge vertices placed at rational points, not all coordinates of the vertices lie in a radical extension of the distance field.
U2 - 10.1090/S0002-9947-06-04049-9
DO - 10.1090/S0002-9947-06-04049-9
M3 - Journal article
VL - 359
SP - 2269
EP - 2303
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 1088-6850
IS - 5
ER -