Saturation Problems on Graphs
Date
2023-08-25
Authors
Ogden, Shannon
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Abstract
In this thesis, we consider two variations on classical saturation problems in extremal graph theory: rainbow saturation and weak saturation.
An edge-coloured graph G is rainbow if every edge in G receives a distinct colour. Given a graph H, an edge-coloured graph G is H-rainbow-saturated if G does not contain a rainbow copy of H, but the addition of any non-edge to G, in any colour, creates a rainbow copy of H. The rainbow saturation number of H, denoted by rsat(n,H), is the minimum number of edges in an H-rainbow saturated graph on n vertices. In Chapter 2, we prove that, like ordinary saturation numbers, the rainbow saturation number of every graph H is linear in n. This result confirms a conjecture of Girao, Lewis, and Popielarz.
In Chapter 3, we consider a specific type of weak saturation known as r-bond bootstrap percolation. In the r-bond bootstrap percolation process on a graph G, we start with a set of initially infected edges of G, and consider all other edges in G to be healthy. At each subsequent step in the process, the infection spreads to a healthy edge if at least one of its endpoints is incident with at least r infected edges. Once an edge is infected, it remains infected indefinitely. If a set of initially infected edges will eventually infect all of E(G), we refer to it as an r-percolating set of G. Define m_e(G,r) to be the minimum number of edges in an r-percolating set of G.
Recently, Hambardzumyan, Hatami, and Qian introduced a clever new polynomial method, which they used to provide recursive formulas for m_e(G,r) when G is either a d-dimensional torus or a d-dimensional grid. We push this polynomial method further, in order to determine m_e(G,r) for certain other graphs G. In particular, we provide recursive formulas for m_e(G,r) when G is a Cartesian product of stars or a Cartesian product of joined cycles (cycles with a single chord). We also give upper and lower bounds on m_e(G,r) when G is a Cartesian product of a tree with any graph H, and examine the conditions under which these bounds match.
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Keywords
Graph Theory, Saturation, Rainbow Saturation, Weak Saturation, Bootstrap Percolation