<p dir="ltr">Combinatorial graph theory is an essential tool in the design and analysis of cryptographic primitives such as Memory-Hard Functions (MHFs) and Proofs of Sequential Work (PoSWs). MHFs are used to design egalitarian Proofs of Work and to help protect low-entropy secrets such as user passwords against brute-force attacks in password hashing. A PoSW is a protocol for proving that one spent significant sequential computation work to validate some statement. PoSWs have many applications, including time-stamping, blockchain design, and universally verifiable CPU benchmarks. Prior work has used combinatorial properties of graphs to construct provably secure MHFs and PoSWs. However, some open problems still exist, such as improving security bounds for MHFs, finding approximation algorithms for measuring their memory hardness, and analyzing the post-quantum security of MHFs and PoSWs. This dissertation addresses these challenges in the security analysis of MHFs and PoSWs using combinatorial graph theory. </p><p dir="ltr">We first improve the understanding of the classical security of MHFs in the following ways. (1) We present improved security bounds for MHF candidates such as Argon2i and DRSample under plausible graph-theoretic conjectures. (2) We prove that it is Unique Games-hard to approximate the cumulative pebbling complexity of a directed acyclic graph, which is an important metric to understand the memory-hardness of data-independent MHFs. (3) We provide the first explicit construction of extremely depth-robust graphs with small indegree. Here, (extreme) depth-robustness is a crucial combinatorial tool to construct secure MHFs and PoSWs. (4) We build a new family of graphs that achieves better provable parameters for concrete depth-robustness.</p><p dir="ltr">Second, as we progress toward developing quantum computers, we initiate the post-quantum security analysis of MHFs and PoSWs. Specifically, we make the following contributions. (1) We introduce the parallel reversible pebbling game, which captures additional restrictions in quantum computing. We use combinatorial graph theory as a tool to analyze the space-time complexity and the cumulative pebbling complexity of MHF candidates such as Argon2i and DRSample in a reversible setting, which we call reversible space-time/cumulative pebbling cost, respectively. (2) We prove that the reversible cumulative pebbling cost is never too much larger than the classical cumulative pebbling cost, along with the separation result that, in some instances, the reversible cumulative pebbling cost is asymptotically larger than the classical one. (3) We prove that it is also Unique Games-hard to approximate the reversible cumulative pebbling cost of a directed acyclic graph. (4) Finally, we establish the post-quantum security of a PoSW from Cohen and Pietrzak (EUROCRYPT 2018) in the parallel quantum random oracle model by extending Zhandry's compressed oracle technique (CRYPTO 2019) and utilizing underlying combinatorial techniques of PoSWs.</p>