Higher-order Van Hove singularities in strongly correlated electron systems provide a fertile ground for emergent electronic orders and superconductivity. This study investigates the interplay between magnetic fluctuations and superconducting pairing near higher-order Van Hove singularities on the honeycomb lattice, a paradigmatic platform relevant to graphene. By incorporating third-nearest-neighbor hopping $t^{\prime \prime }$, we uncover a universal crossover: ferromagnetic fluctuations dominate below the higher-order Van Hove filling, while antiferromagnetic fluctuations take over toward half filling. A key finding is that the already dominant $f_n$-wave pairing is enhanced in the critical region of this magnetic crossover by the higher-order Van Hove. This enhancement is driven by the synergistic effect of the higher-order Van Hove singularities-induced divergent density of states and the competing magnetic fluctuations. Although increased hopping parameters generally suppress superconducting correlation, we identify a critical $t^{\prime \prime }$ that anomalously enhances pairing via the higher-order Van Hove renormalization. Furthermore, the nearest-neighbor Coulomb interaction suppresses the pairing correlation function in a sign-independent manner. Our results clarify the competitive mechanisms between magnetic fluctuations and unconventional superconductivity in higher-order Van Hove singularities systems, offering a theoretical basis for tailoring quantum phases in graphene-based materials via band engineering.

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• Pairing mechanisms
• Superconductivity
• Van Hove singularity
• Honeycomb lattice
• Diffusion quantum Monte Carlo
• Hubbard model
• Monte Carlo methods