Sufficient degree conditions for graph embeddings

In this dissertation, we focus on the sufficient conditions to guarantee one graph being the subgraph of another. In Chapter 2, we discuss list packing, a modification of the idea of graph packing. This is fitting one graph in the complement of another graph. Sauer and Spencer showed a sufficient bound involving maximum degrees, and this was further explored by Kaul and Kostochka to characterize all extremal cases. Bollobas and Eldridge (and independently Sauer and Spencer) developed edge sum bounds to guarantee packing. In Chapter 2, we introduce the new idea of list packing and use it to prove stronger versions of many existing theorems. Namely, for two graphs, if the product of the maximum degrees is small or if the total number of edges is small, then the graphs pack. In Chapter 3, we discuss the problem of finding k vertex-disjoint cycles in a multigraph. This problem originated from a conjecture of Erdos and has led to many different results. Corradi and Hajnal looked at a minimum degree condition. Enomoto and Wang independently looked at a minimum degree-sum condition. More recently, Kierstead, Kostochka, and Yeager characterized the extremal cases to improve these bounds. In Chapter 3, we improve on the multigraph degree-sum result. We characterize all multigraphs that have simple Ore-degree at least 4k -3 , but do not contain k vertex-disjoint cycles. Moreover, we provide a polynomial time algorithm for deciding if a graph contains k vertex-disjoint cycles. Lastly, in Chapter 4, we consider the same problem but with chorded cycles. Finkel looked at the minimum degree condition while Chiba, Fujita, Gao, and Li addressed the degree-sum condition. More recently, Molla, Santana, and Yeager improved this degree-sum result, and in Chapter 4, we will improve on this further

Parameterized Directed kk-Chinese Postman Problem and kk Arc-Disjoint Cycles Problem on Euler Digraphs

In the Directed $k$-Chinese Postman Problem ($k$-DCPP), we are given a connected weighted digraph $G$ and asked to find $k$ non-empty closed directed walks covering all arcs of $G$ such that the total weight of the walks is minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128) asked for the parameterized complexity of $k$-DCPP when $k$ is the parameter. We prove that the $k$-DCPP is fixed-parameter tractable. We also consider a related problem of finding $k$ arc-disjoint directed cycles in an Euler digraph, parameterized by $k$. Slivkins (ESA 2003) showed that this problem is W[1]-hard for general digraphs. Generalizing another result by Slivkins, we prove that the problem is fixed-parameter tractable for Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler digraphs remains W[1]-hard even for Euler digraphs

Edge-disjoint Hamilton cycles in graphs

In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree \delta must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every \alpha > 0, every (almost) regular and sufficiently large graph on n vertices with minimum degree at least $(1/2 + \alpha)n$ can be almost decomposed into edge-disjoint Hamilton cycles.Comment: Minor Revisio