Abstract
We observe recent results on the applications of extremal graph theory to cryptography. Classical Extremal Graph Theory contains Erdős Even Circuite Theorem and other remarkable results on the maximal size of graphs without certain cycles. Finite automaton is roughly a directed graph with labels on directed arrows. The most important advantage of Turing machine in comparison with finite automaton is existence of “potentially infinite memory”. In terms of Finite Automata Theory Turing machine is an infinite sequence of directed graphs with colours on arrows. This is a motivation of studies of infinite families of extremal directed graphs without certain commutative diagrams. The explicite constructions of simple and directed graphs of large girth (or large cycle indicator) corresponds to efficient encryption of Turing machines.
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