Strong and Hiding Distributed Certification of Bipartiteness

In this paper, we study the problem of certifying whether a graph is bipartite (i.e. $2$-colorable) with a locally checkable proof (LCP) that is able to hide a $2$-coloring from the verifier. More precisely, we say an LCP for $2$-coloring is hiding if, in a yes-instance, it is possible to assign certificates to nodes without revealing an explicit $2$-coloring. Motivated by the search for promise-free separations of extensions of the LOCAL model in the context of locally checkable labeling (LCL) problems, we also require the LCPs to satisfy what we refer to as the strong soundness property. This is a strengthening of soundness that requires that, in a no-instance (i.e., a non-$2$-colorable graph) and for every certificate assignment, the subset of accepting nodes must induce a $2$-colorable subgraph. We show that strong and hiding LCPs for $2$-coloring exist in specific graph classes and requiring only $O(\log n)$-sized certificates. Furthermore, when the input is promised to be a cycle or contains a node of degree $1$, we show the existence of strong and hiding LCPs even in an anonymous network and with constant-size certificates. Despite these upper bounds, we prove that there are no strong and hiding LCPs for $2$-coloring in general, unless the algorithm has access to node identifiers and uses certificates of size~$\omega(1)$. Furthermore, in anonymous networks, the lower bound holds regardless of the certificate size. The proof relies on a Ramsey-type result as well as an argument about the realizability of subgraphs of the neighborhood graph consisting of the accepting views of an LCP. Along the way, we also give a characterization of the hiding property for the general $k$-coloring problem that appears to be a key component for future investigations in this context.