Zero-shot link prediction (ZSLP) on knowledge graphs aims at automatically identifying relations between given entities. Existing methods primarily employ auxiliary information to predict tail entity given head entity and its relation, yet face challenges due to the occasional unavailability of such detailed information and the inherent simplicity of predicting tail entities based on semantic similarities. Even though Large Language Models (LLMs) offer a promising solution to predict unobserved relations between the head and tail entity in a zero-shot manner, their performance is still restricted due to the inability to leverage all the (exponentially many) paths' information between two entities, which are critical in collectively indicating their relation types. To address this, in this work, we introduce a Condensed Transition Graph Framework for Zero-Shot Link Prediction (CTLP), which encodes all the paths' information in linear time complexity to predict unseen relations between entities, attaining both efficiency and information preservation. Specifically, we design a condensed transition graph encoder with theoretical guarantees on its coverage, expressiveness, and efficiency. It is learned by a transition graph contrastive learning strategy. Subsequently, we design a soft instruction tuning to learn and map the all-path embedding to the input of LLMs. Experimental results show that our proposed CTLP method achieves state-of-the-art performance on three standard ZSLP datasets