Fault-tolerant connectivity labelings are schemes that, given an $n$-vertex graph $G=(V,E)$ and $f\geq 1$, produce succinct yet informative labels for the elements of the graph. Given only the labels of two vertices $u,v$ and of the elements in a faulty-set $F$ with $|F|\leq f$, one can determine if $u,v$ are connected in $G-F$, the surviving graph after removing $F$. For the edge or vertex faults models, i.e., $F\subseteq E$ or $F\subseteq V$, a sequence of recent work established schemes with $poly(f,\log n)$-bit labels. This paper considers the color faults model, recently introduced in the context of spanners [Petruschka, Sapir and Tzalik, ITCS'24], which accounts for known correlations between failures. Here, the edges (or vertices) of the input $G$ are arbitrarily colored, and the faulty elements in $F$ are colors; a failing color causes all edges (vertices) of that color to crash. Our main contribution is settling the label length complexity for connectivity under one color fault ($f=1$). The existing implicit solution, by applying the state-of-the-art scheme for edge faults of [Dory and Parter, PODC'21], might yield labels of $\Omega(n)$ bits. We provide a deterministic scheme with labels of $\tilde{O}(\sqrt{n})$ bits in the worst case, and a matching lower bound. Moreover, our scheme is universally optimal: even schemes tailored to handle only colorings of one specific graph topology cannot produce asymptotically smaller labels. We extend our labeling approach to yield a routing scheme avoiding a single forbidden color. We also consider the centralized setting, and show an $\tilde{O}(n)$-space oracle, answering connectivity queries under one color fault in $\tilde{O}(1)$ time. Turning to $f\geq 2$ color faults, we give a randomized labeling scheme with $\tilde{O}(n^{1-1/2^f})$-bit labels, along with a lower bound of $\Omega(n^{1-1/(f+1)})$ bits.

翻译：容错连通性标记方案允许对给定$n$顶点图$G=(V,E)$和参数$f\geq 1$，为图元素生成简洁而富有信息的标签。仅凭两个顶点$u,v$的标签以及故障集$F$（满足$|F|\leq f$）中元素的标签，即可判断在移除$F$后的存留图$G-F$中$u,v$是否连通。针对边故障或顶点故障模型（即$F\subseteq E$或$F\subseteq V$），近期一系列工作已建立标签长度为$poly(f,\log n)$比特的方案。本文考虑颜色故障模型——该模型最近在spanner场景中被引入[Petruschka, Sapir and Tzalik, ITCS'24]，用于建模故障间的已知相关性。在该模型中，输入图$G$的边（或顶点）被任意着色，故障集$F$中的元素为颜色；一种颜色失效会导致所有该颜色的边（或顶点）崩溃。我们的主要贡献在于解决了单颜色故障（$f=1$）下的连通性标签长度复杂度问题。现有隐式方案（通过应用[Dory and Parter, PODC'21]针对边故障的最优方案）可能导致$\Omega(n)$比特的标签。我们提出一种确定性方案，最坏情况下标签长度为$\tilde{O}(\sqrt{n})$比特，并给出匹配的下界。此外，该方案具有全局最优性：即使专门针对某特定图拓扑的着色定制的方案，也无法产生渐近更小的标签。我们将标签方法扩展至规避单种禁止颜色的路由方案。我们还考虑了集中式场景，给出一个$\tilde{O}(n)$空间规模的预言机，可在$\tilde{O}(1)$时间内回答单颜色故障下的连通性查询。针对$f\geq 2$颜色故障，我们提出一种随机化标记方案，标签长度为$\tilde{O}(n^{1-1/2^f})$比特，并给出了$\Omega(n^{1-1/(f+1)})$比特的下界。