| Summary: | A major theme in modern graph theory is the exploration of maximal packings and minimal covers of graphs with subgraphs in some given family. We focus on packings and coverings with Hamilton cycles, and prove the following results in the area. • Let ε > 0, and let \(G\) be a large graph on n vertices with minimum degree at least (1=2+ ε)n. We give a tight lower bound on the size of a maximal packing of \(G\) with edge-disjoint Hamilton cycles. • Let \(T\) be a strongly k-connected tournament. We give an almost tight lower bound on the size of a maximal packing of \(T\) with edge-disjoint Hamilton cycles. • Let log \(^1\)\(^1\)\(^7\) \(n\)/\(n\)≤\(p\)≤1-\(n\)\(^-\)\(^1\)\(^/\)\(^8\). We prove that \(G\)\(_n\)\(_,\)\(_p\) may a.a.s be covered by a set of ⌈Δ(\(G\)\(_n\)\(_,\)\(_p\))/2⌉ Hamilton cycles, which is clearly best possible. In addition, we consider some problems in on-line Ramsey theory. Let r(\(G\),\(H\)) denote the on-line Ramsey number of \(G\) and \(H\). We conjecture the exact values of r (\(P\)\(_k\),\(P\)\(_ℓ\)) for all \(k\)≤ℓ. We prove this conjecture for \(k\)=2, prove it to within an additive error of 10 for \(k\)=3, and prove an asymptotically tight lower bound for \(k\)=4. We also determine r(\(P\)\(_3\),\(C\)\(_ℓ\) exactly for all ℓ.
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