In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every α > 0, every sufficiently large graph on n vertices with minimum degree at least(1/2 + α)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 δ must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every α > 0, every (almost) regular and sufficiently large graph on n vertices with minimum degree at least (1/2 + α)n can be almost decomposed into edge-disjoint Hamilton cycles.

Riis [Electron. J. Combin., 14(1):R44, 2007] introduced a guessing game for graphs which is equivalent to finding protocols for network coding. In this paper we prove upper and lower bounds for the winning probability of the guessing game on undirected graphs. We find optimal bounds for perfect graphs and minimally imperfect graphs, and present a conjecture relating the exact value for all graphs to the fractional chromatic number.

ABSTRACT:The Alon-Roichman theorem states that for every ε > 0 there is a constant c(ε), such that the Cayley graph of a finite group G with respect to c(ε) log |G| elements of G, chosen independently and uniformly at random, has expected second largest eigenvalue less than ε. In particular, such a graph is an expander with high probability.Landau and Russell, and independently Loh and Schulman, improved the bounds of the theorem. Following Landau and Russell we give a new proof of the result, improving the bounds even further. When considered for a general group G, our bounds are in a sense best possible. We also give a generalization of the Alon-Roichman theorem to random coset graphs.Our proof uses a Hoeffding-type result for operator valued random variables, which we believe can be of independent interest.

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We provide an NC algorithm for finding Hamilton cycles in directed graphs with a certain robust expansion property. This property captures several known criteria for the existence of Hamilton cycles in terms of the degree sequence and thus we provide algorithmic proofs of (i) an 'oriented' analogue of Dirac's theorem and (ii) an approximate version (for directed graphs) of Chvátal's theorem. Moreover, our main result is used as a tool in a recent paper by Kühn and Osthus, which shows that regular directed graphs of linear degree satisfying the above robust expansion property have a Hamilton decomposition, which in turn has applications to TSP tour domination.