Hereditary Extended Properties, Quasi-Random Graphs and Induced Subgraphs

• Authors: Miklós Simonovits;Vera T. Sós
• Venue: Combinatorics, Probability and Computing
• Year: 2003

This is a continuation of our work on quasi-random graph properties. The class of quasi-random graphs is defined by certain equivalent graph properties possessed by random graphs. One of the most important of these properties is that, for fixed $\nu$, every fixed sample graph $L_\nu$ has the same frequency in $G_n$ as in the $p$-random graph. (This holds for both induced and not necessarily induced containment.) In B;9D; we proved that, if the frequency of just one fixed $L_\nu$ – as a not necessarily induced subgraph – in every ‘large’ induced subgraph $F_h\subseteq G_n$ is the same as for the random graphs, then $(G_n)$ is quasi-random. Here we shall investigate the analogous problem for induced subgraphs $L_\nu$. In such cases $(G_n)$ is not necessarily quasi-random.We shall prove, among other things, that, for every regular sample graph $L_\nu$, $\nu\geqslant 4$, if the number of induced copies of $L_\nu$ in every induced $F_h\subseteq G_n$ is asymptotically the same as in a $p$-random graph (up to an error term $o(n^\nu)$), then $(G_n)$ is the union of (at most) two quasi-random graph sequences, with possibly distinct attached probabilities (assuming that $p\in (0,1)$, $e(L_\nu)0$, and $L_\nu\ne K_\nu$).We conjecture the same conclusion for every $L_\nu$ with $\nu\ge 4$, i.e., even if we drop the assumption of regularity.We shall reduce the general problem to solving a system of polynomials. This gives a ‘simple’ algorithm to decide the problem for every given $L_\nu$.