The ``arms race'' of Large Language Models (LLMs) demands new benchmarks to examine their progresses. In this paper, we introduce GraphArena, a benchmarking tool designed to evaluate LLMs on real-world graph computational problems. It offers a suite of four polynomial-time tasks (e.g., Shortest Distance) and six NP-complete challenges (e.g., Traveling Salesman Problem). GraphArena features a rigorous evaluation framework that classifies LLM outputs as correct, suboptimal (feasible but not optimal), hallucinatory (properly formatted but infeasible), or missing. Evaluation of over 10 LLMs reveals that even top-performing LLMs struggle with larger, more complex graph problems and exhibit hallucination issues. We further explore four potential solutions to address this issue and improve LLMs on graph computation, including chain-of-thought prompting, instruction tuning, code writing, and scaling test-time compute, each demonstrating unique strengths and limitations. GraphArena complements the existing LLM benchmarks and is open-sourced at https://github.com/squareRoot3/GraphArena.

The k-core, which refers to the induced subgraph with a minimum degree of at least k, is widely used in cohesive subgraph discovery and has various applications. However, the k-core in real-world graphs tends to be extremely large, which hinders its effectiveness in practical applications. This challenge has motivated researchers to explore a variant of the k-core problem known as the minimum k-core search problem. This problem has been proven to be NP-Hard, and most of the existing studies naturally either deal with approximate solutions or suffer from inefficiency in practice. In this paper, we focus on designing efficient exact algorithms for the minimum k-core search problem. In particular, we develop an iterative-based framework that decomposes an instance of the minimum k-core search problem into a list of problem instances on another well-structured graph pattern. Based on this framework, we propose an iterative-based branch-and-bound algorithm, namely IBB, with additional pruning and reduction techniques. We show that, with a n-vertex graph, IBB runs in cn nO(1) time for some c < 2, achieving better theoretical performance than the trivial bound of 2n nO(1). Finally, our experiments on real-world graphs demonstrate that IBB is up to three orders of magnitude faster than the state-of-the-art algorithms on real-world datasets.