The field of graph theory is witnessing significant developments in the study of Hamiltonian paths and cycles, with a focus on understanding the complexity of these problems in various graph classes. Researchers are exploring the relationships between graph width parameters, such as treewidth and pathwidth, and the complexity of finding Hamiltonian paths and cycles. Additionally, there is a growing interest in graph layouts, including stack, queue, and priority queue layouts, which have applications in graph visualization and parsing. The introduction of priority queues has led to new characterization results and recognition algorithms for graphs that admit priority queue layouts. Furthermore, verification of conjectures, such as the BHR Conjecture, is being pursued using a combination of mathematical strategies and computational methods. Notable papers in this area include:

• A Graph Width Perspective on Partially Ordered Hamiltonian Paths and Cycles I, which provides new complexity results for Hamiltonian paths and cycles in graphs of bounded pathwidth and treewidth.
• Verification of Hamiltonian Path Conjecture (BHR Conjecture) for Integers up to p=31, which presents an approach to the conjecture based on frequency partitions and local/global adjustment operations and backtracking, and reports experimental evidence for primes up to 31.