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{\bf Vadim V. Lozin, Colin Mayhill and Victor Zamaraev}
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{\bf A Note on the Speed of Hereditary Graph Properties}
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For a graph property $X$, let $X_n$ be the number of graphs with vertex set $\{1,\ldots,n\}$
having property $X$, also known as the speed of $X$. A property $X$ is called factorial if $X$ is
hereditary (i.e. closed under taking induced subgraphs) and $n^{c_1n}\le X_n\le n^{c_2n}$ for some
positive constants $c_1$ and $c_2$. Hereditary properties with the speed slower than factorial are
surprisingly well structured.
The situation with factorial properties is more complicated and less explored, although
this family includes many properties of theoretical or practical importance,
such as planar graphs or graphs of bounded vertex degree.
To simplify the study of factorial properties, we propose the following conjecture:
the speed of a hereditary property $X$ is factorial if and only if the fastest of
the following three properties is factorial: bipartite graphs in $X$, co-bipartite graphs
in $X$ and split graphs in $X$. In this note, we verify the conjecture for hereditary properties defined
by forbidden induced subgraphs with at most 4 vertices.
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