Brief Announcement: Strong and Hiding Distributed Certification of $k$-Coloring

We study the problem of certifying whether a graph is k-colorable with a locally checkable proof (LCP) that is able to hide the k-coloring from the verifier, in the sense that no algorithm can (completely) extract a k-coloring from the certificate. 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 call the strong soundness property. We focus on the case of 2-coloring and show that strong and hiding LCPs for 2-coloring exist in specific graph classes and require 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, regardless of certificate size. Along the way, we 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.