Minimal Universal Bipartite Graphs Vadim V. Lozin, Gábor Rudolf RUTCOR – Rutgers University

Hereditary classes

Universal Graphs Trivial bounds:

Optimal universal graphs A sequence of universal X-graphs is asimptotically optimal if optimal in order if

Index of a hereditary class V.E. Alekseev (1992):

Unitary classes Graph classes of the same index constitute a stratum Classes of index 1: unitary stratum Contains many interesting classes: Forests Planar graphs Interval graphs Permutation graphs Line graphs Etc.

Isometry of graph classes The equivalence classes are called layers Scheinarman and Zito (1994): The first four layers in the unitary stratum are:

Exponential classes Alexeev (1997) For each exponential class X there is a constant p such that every graph G in X can be partitioned into at most p subsets each of which is either an independent set or a clique and between any two subsets there are either all possible edges or none of them. Asimptotically optimal universal graphs exist

Minimal factorial classes of bipartite graphs Alexeev (1997) The three minimal factorial classes of bipartite graphs are: Asimptotically optimal universal graphs exist for all these classes (trivial for ).

Canonical graphs

Universal chain graphs is an n-universal chain graph

Universal chain graphs is an n-universal chain graph

Universal chain graphs is an n-universal chain graph

Universal biparite permutation graphs Brandtstädt, Lozin (2003): is an n-universal bipartite permutation graph; it is optimal in order

Unit interval graphs

General bipartite graphs

Let us first consider :

Optimality