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