Inductive definition of two restricted classes of triangulations

AbstractThe inductive definitions of (i) the class of all triangulations (of the sphere) without vertices of degree 3; and (ii) the class of all triangulations with all vertices of even degree are given. The dual rules give us (i) the class of all 3-connected planar cubic graphs without triangles; and (ii) the class of all 3-connected bipartite planar cubic graphs (related to Barnette's hamiltonicity conjecture)

Construction of planar 4-connected triangulations

In this article we describe a recursive structure for the class of 4-connected triangulations or - equivalently - cyclically 4-connected plane cubic graphs

One brick at a time: a survey of inductive constructions in rigidity theory

We present a survey of results concerning the use of inductive constructions to study the rigidity of frameworks. By inductive constructions we mean simple graph moves which can be shown to preserve the rigidity of the corresponding framework. We describe a number of cases in which characterisations of rigidity were proved by inductive constructions. That is, by identifying recursive operations that preserved rigidity and proving that these operations were sufficient to generate all such frameworks. We also outline the use of inductive constructions in some recent areas of particularly active interest, namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar frameworks. We summarize the key outstanding open problems related to inductions.Comment: 24 pages, 12 figures, final versio

A new Kempe invariant and the (non)-ergodicity of the Wang-Swendsen-Kotecky algorithm

We prove that for the class of three-colorable triangulations of a closed oriented surface, the degree of a four-coloring modulo 12 is an invariant under Kempe changes. We use this general result to prove that for all triangulations T(3L,3M) of the torus with 3<= L <= M, there are at least two Kempe equivalence classes. This result implies in particular that the Wang-Swendsen-Kotecky algorithm for the zero-temperature 4-state Potts antiferromagnet on these triangulations T(3L,3M) of the torus is not ergodic.Comment: 37 pages (LaTeX2e). Includes tex file and 3 additional style files. The tex file includes 14 figures using pstricks.sty. Minor changes. Version published in J. Phys.