A graph $G$ is called self-ordered (a.k.a asymmetric) if the identity
permutation is its only automorphism. Equivalently, there is a unique
isomorphism from $G$ to any graph that is isomorphic to $G$. We say that
$G=(V,E)$ is robustly self-ordered if the size of the symmetric difference
between $E$ and the edge-set of the graph obtained by permuting $V$ using any
permutation $\pi:V\to V$ is proportional to the number of non-fixed-points of
$\pi$. In this work, we initiate the study of the structure, construction and
utility of robustly self-ordered graphs.
We show that robustly self-ordered bounded-degree graphs exist (in
abundance), and that they can be constructed efficiently, in a strong sense.
Specifically, given the index of a vertex in such a graph, it is possible to
find all its neighbors in polynomial-time (i.e., in time that is
poly-logarithmic in the size of the graph).
We also consider graphs of unbounded degree, seeking correspondingly
unbounded robustness parameters. We again demonstrate that such graphs (of
linear degree) exist (in abundance), and that they can be constructed
efficiently, in a strong sense. This turns out to require very different tools.
Specifically, we show that the construction of such graphs reduces to the
construction of non-malleable two-source extractors (with very weak parameters
but with some additional natural features).
We demonstrate that robustly self-ordered bounded-degree graphs are useful
towards obtaining lower bounds on […]

We study the complexity of proving that a sparse random regular graph on an
odd number of vertices does not have a perfect matching, and related problems
involving each vertex being matched some pre-specified number of times. We show
that this requires proofs of degree $\Omega(n / \log n)$ in the Polynomial
Calculus (over fields of characteristic $\ne 2$) and Sum-of-Squares proof
systems, and exponential size in the bounded-depth Frege proof system. This
resolves a question by Razborov asking whether the LovĂˇsz-Schrijver proof
system requires $n^\delta$ rounds to refute these formulas for some $\delta >
0$. The results are obtained by a worst-case to average-case reduction of these
formulas relying on a topological embedding theorem which may be of independent
interest.