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On representable graphs

Kitaev, Sergey and Pyatkin, Artem (2008) On representable graphs. Journal of Automata, Languages and Combinatorics, 13 (1). pp. 45-54.

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Abstract

A graph G = (V;E) is representable if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if (x; y) 2 E for each x 6= y. If W is k-uniform (each letter of W occurs exactly k times in it) then G is called k-representable. We prove that a graph is representable if and only if it is k-representable for some k. Examples of non-representable graphs are found in this paper. Some wide classes of graphs are proven to be 2- and 3-representable. Several open problems are stated.