Sachal Abdullah

I explored Sheehan's Conjecture on the non-existence of uniquely Hamiltonian regular graphs. In an attempt to produce an infinite family of uniquely Hamiltonian 4-regular graphs from cubic hypohamiltonian graphs — following a construction due to Fleischner — I showed that in general this construction does not hold for all cubic hypohamiltonian graphs except for the Petersen graph. I demonstrated the limitations of Fleischner's construction by constructing examples via the 1st and 2nd Blanuša snarks. To continue the development of the theory of uniquely Hamiltonian graphs, I formulated a number of open problems which arose from this research. In preparation for a preprint publication.

Supervisor: Dr. Andrew Goodall — Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague

Conducted foundational literature reviews and exploratory investigations on uniquely Hamiltonian graphs and the Hadwiger–Debrunner \((p,q)\)-problem. Surveyed the probabilistic method and analysed the Hamiltonian cycle property for bipartite and regular graphs — establishing the theoretical groundwork for subsequent research on Sheehan's Conjecture.

Supervisor: Dr. Mudassir Shabbir — Information Technology University, Lahore

Undergraduate thesis: developed an automated, dynamic traffic-signal control system to mitigate urban congestion using computer vision, game theory, and reinforcement learning. The system uses background subtraction with Gaussian distributions to extract real-time vehicle density and velocity from video feeds, then models intersecting roads as competing players within a Markov framework — applying Nash Equilibrium and minimax strategies to optimize green-light durations under live traffic conditions.

Supervisor: Dr. Nazim Ashraf — Department of Computer Science, Forman Christian College, Lahore