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Universal graphs and universal permutations

Atminas, Aistis and Kitaev, Sergey and Lozin, Vadim and Valyuzhenich, Alexander (2013) Universal graphs and universal permutations. Discrete Mathematics, Algorithms and Applications, 5 (4). ISSN 1793-8317

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Let X be a family of graphs and Xn the set of n-vertex graphs in X. A graph U(n) containing all graphs from Xn as induced subgraphs is called n-universal for X. Moreover, we say that U(n) is a propern-universal graph for X if it belongs to X. In the present paper, we construct a proper n-universal graph for the class of split permutation graphs. Our solution includes two ingredients: a proper universal 321-avoiding permutation and a bijection between 321-avoiding permutations and symmetric split permutation graphs. The n-universal split permutation graph constructed in this paper has 4n3 vertices, which means that this construction is order-optimal.