Multi-label topic classification without labeled training data is a challenging task, specially when documents contain complex relational information. We present a zero-shot multi-label topic classification framework and systematically investigate how per-article knowledge graph augmentation affects its performance. The base framework classifies topics in documents without labeled training data and has four variants: article-only classification, keyword-enhanced classification, and self-consistency decoding variants of both. Then, we augment each base variant with per article knowledge graph. This graph is extracted from the input document through a pipeline similar to KGGen based on subject-predicate-object triples. We test all eight methods, four base and four graph augmented on fifteen LLMs and eight multi-label datasets across different domains. For the base framework, keyword-enhanced classification (AK) is the best performing method, and six out of fifteen LLMs surpass the sentence-encoder baseline. Graph augmentation has positive and negative impacts on small and large models, respectively. This shows that larger models already contain enough relational information from pretraining. Furthermore, the self-consistency decoding variant does not show performance improvements in any experiment while increasing computation costs about fivefold.