A locally checkable proof (LCP) is a non-deterministic distributed algorithm designed to verify global properties of a graph $G$. It involves two key components: a prover and a distributed verifier. The prover is an all-powerful computational entity capable of performing any Turing-computable operation instantaneously. Its role is to convince the distributed verifier -- composed of the graph's nodes -- that $G$ satisfies a particular property $\Pi$. We study the problem of certifying whether a graph is $k$-colorable with an LCP that is able to hide the $k$-coloring from the verifier. More precisely, we say an LCP for $k$-coloring is hiding if, in a yes-instance, it is possible to assign certificates to nodes without revealing an explicit $k$-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-$k$-colorable graph) and for every certificate assignment, the subset of accepting nodes must induce a $k$-colorable subgraph. We focus on the case of $2$-coloring. 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, regardless of certificate size. The proof relies on a Ramsey-type result as well as an intricate argument about the realizability of subgraphs of the neighborhood graph consisting of the accepting views of an LCP.

翻译：局部可检查证明（LCP）是一种用于验证图$G$全局性质的非确定性分布式算法。它包含两个关键组成部分：证明者和分布式验证者。证明者是一个全能的计算实体，能够瞬时执行任何图灵可计算的操作。其作用是使由图的节点组成的分布式验证者确信$G$满足特定性质$\Pi$。我们研究了用LCP认证图是否可$k$着色的问题，并要求该LCP能够对验证者隐藏具体的$k$着色方案。更精确地说，如果在一个是实例中，可以为节点分配证书而不暴露明确的$k$着色方案，则称该$k$着色LCP是隐藏的。受在局部可检查标记（LCL）问题背景下寻求LOCAL模型扩展的无承诺分离的动机驱动，我们还要求LCP满足我们称之为强可靠性性质的特性。这是对可靠性的强化，要求在否实例（即不可$k$着色的图）中，对于任何证书分配，接受节点构成的子图必须是可$k$着色的。我们重点关注$2$着色的情况。我们证明了在特定图类中存在强隐藏的$2$着色LCP，且仅需$O(\log n)$大小的证书。此外，当输入被承诺是一个环或包含一个度为$1$的节点时，我们证明了即使在匿名网络中，也存在具有常数大小证书的强隐藏LCP。尽管存在这些上界，我们证明了在一般情况下，无论证书大小如何，都不存在强隐藏的$2$着色LCP。该证明依赖于一个拉姆齐型结果，以及关于由LCP的接受视图构成的邻域图的子图可实现性的复杂论证。