Irreducible triangulations of the Möbius band

A complete list of irreducible triangulations is identified on the Möbius band.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Ciencia e Innovació

The geometry of flip graphs and mapping class groups

The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a sort of combinatorial Teichm\"uller space and is quasi-isometric to the underlying mapping class group. We study this space in two main directions. We first show that strata corresponding to triangulations containing a same multiarc are strongly convex within the whole space and use this result to deduce properties about the mapping class group. We then focus on the quotient of this space by the mapping class group to obtain a type of combinatorial moduli space. In particular, we are able to identity how the diameters of the resulting spaces grow in terms of the complexity of the underlying surfaces.Comment: 46 pages, 23 figure

Some results on cubic graphs

Pursuing a question of Oxley, we investigate whether the edge set of a graph admits a bipartition so that the contraction of either partite set produces a series-parallel graph. While Oxley\u27s question in general remains unanswered, our investigations led to two graph operations (Chapters 2 and 4) which are of independent interest. We present some partial results toward Oxley\u27s question in Chapter 3. The central results of the dissertation involve an operation on cubic graphs called the switch; in the literature, a similar operation is known as the edge slide. In Chapter 2, the author proves that we can transform, with switches, any connected, cubic graph on n vertices into any other connected, cubic graph on n vertices. Furthermore, connectivity, up to internal 4-connectedness, can be preserved during the operations. In 2007, Demaine, Hajiaghayi, and Mohar proved the following: for a fixed genus g and any integer k greater than or equal to 2, and for every graph G of Euler genus at most g, the edges of G can be partitioned into k sets such that contracting any one of the sets produces a graph of tree-width at most O(g^2 k). In Chapter 3 we sharpen this result, when k=2, for the projective plane (g=1) and the torus (g=2). During early simultaneous investigations of Jaeger\u27s Dual-Hamiltonian conjecture and Oxley\u27s question, we obtained a simple structure theorem on cubic, internally 4-connected graphs. That result is found in Chapter 4