Thin Trees in Some Families of Graphs

Abstract

Let 𝐺=(𝑉,𝐸) be a graph and let 𝑇 be a spanning tree of 𝐺. The thinness parameter of 𝑇 denoted by 𝜌(𝑇) is the maximum over all cuts of the proportion of the edges of 𝑇 in the cut. Thin trees play an important role in some recent papers on the Asymmetric Traveling Salesman Problem (ATSP). Goddyn conjectured that every graph of sufficiently large edge-connectivity has a spanning tree 𝑇 such that 𝜌(𝑇) ≤ 𝜀. In this thesis, we study the problem of finding thin spanning trees in two families of graphs, namely, (1) distance-regular graphs (DRGs), and (2) planar graphs.

For some families of DRGs such as strongly regular graphs, Johnson graphs, Crown graphs, and Hamming graphs, we give a polynomial-time construction of spanning trees 𝑇 of maximum degree ≤ 3 such that 𝜌(𝑇) is determined by the parameters of the graph.

For planar graphs, we improve the analysis of Merker and Postle ("Bounded Diameter Arboricity", arXiv:1608.05352v1) and show that every 6-edge-connected planar graph has two edge-disjoint spanning trees 𝑇,𝑇′ such that 𝜌(𝑇),𝜌(𝑇′) ≤ 14⁄15. For 8-edge-connected planar graphs 𝐺, we present a simplified version of the techniques of Merker and Postle and show that 𝐺 has two edge-disjoint spanning trees 𝑇,𝑇′ such that 𝜌(𝑇),𝜌(𝑇′) ≤ 12⁄13.