We study decision problems related to graph properties from the perspective of nondeterministic distributed algorithms. For a yes-instance there must exist a proof that can be verified with a distributed algorithm: all nodes must accept a valid proof, and at least one node must reject an invalid proof. We focus on locally checkable proofs that can be verified with a constant-time distributed algorithm. For example, it is easy to prove that a graph is bipartite: the locally checkable proof gives a $2$-coloring of the graph, which only takes $1$ bit per node. However, it is more difficult to prove that a graph is not bipartite—it turns out that any locally checkable proof requires $\mathrm{\Omega }(\log n)$ bits per node.

In this paper we classify graph properties according to their local proof complexity, i.e., how many bits per node are needed in a locally checkable proof. We establish tight or near-tight results for classical graph properties such as the chromatic number. We show that the local proof complexities form a natural hierarchy of complexity classes: for many classical graph properties, the local proof complexity is either $0$, $\mathrm{\Theta }(1)$, $\mathrm{\Theta }(\log n)$, or $\mathrm{poly}(n)$ bits per node. Among the most difficult graph properties are proving that a graph is symmetric (has a non-trivial automorphism), which requires $\mathrm{\Omega }(n^2)$ bits per node, and proving that a graph is not $3$-colorable, which requires $\mathrm{\Omega }(n^2/\log n)$ bits per node. Any property of connected graphs admits a trivial proof with $O(n^2)$ bits per node.