3 research outputs found

    Packing and embedding large subgraphs

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    This thesis contains several embedding results for graphs in both random and non random settings. Most notably, we resolve a long standing conjecture that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals 1/21/2. %posed e.g.~by Bollob\'as, In Chapter 2 we obtain the following perturbation result regarding the hypercube \cQ^n: if H\subseteq\cQ^n satisfies δ(H)≥αn\delta(H)\geq\alpha n with α>0\alpha>0 fixed and we consider a random binomial subgraph \cQ^n_p of \cQ^n with p∈(0,1]p\in(0,1] fixed, then with high probability H\cup\cQ^n_p contains kk edge-disjoint Hamilton cycles, for any fixed k∈Nk\in\mathbb{N}. This result is part of a larger volume of work where we also prove the corresponding hitting time result for Hamiltonicity. In Chapter 3 we move to a non random setting. %to a deterministic one. %Instead of embedding a single Hamilton cycle our result concerns packing more general families of graphs into a fixed host graph. Rather than pack a small number of Hamilton cycles into a fixed host graph, our aim is to achieve optimally sized packings of more general families of graphs. More specifically, we provide a degree condition on a regular nn-vertex graph GG which ensures the existence of a near optimal packing of any family H\mathcal H of bounded degree nn-vertex kk-chromatic separable graphs into GG. %In general, this degree condition is best possible. %In particular, this yields an approximate version of the tree packing conjecture %in the setting of regular host graphs GG of high degree. %Similarly, our result implies approximate versions of the Oberwolfach problem, %the Alspach problem and the existence of resolvable designs in the setting of %regular host graphs of high degree. In particular, this yields approximate versions of the the tree packing conjecture, the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree

    Sufficient degree conditions for graph embeddings

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    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
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