Abstract
Quasi-random graph properties form a large equivalence class of graph properties which are all shared by random graphs. In recent years, various aspects of these properties have been treated by a number of authors (e.g., see [5]-[14], [16], [23]-[27]). Almost all of these results deal with dense graphs, that is, graphs on n vertices having cn edges for some c > 0 as n → ∞. In this paper, we extend our study of quasi-randomness to sparse graphs, i.e., graphs on n vertices with o(n) edges. It will be shown that many of the quasi-random properties for dense graphs have analogues for sparse graphs, while others do not, at least not without additional hypotheses. In general, sparse graphs are more difficult to deal with than dense graphs, due for example to the possible absence of certain local structures, such as 4-cycles.
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