Abstract:
Lov´asz had posed a question stating whether every connected, vertex-transitive graph
has a Hamilton path in 1969. There is a growing interest in solving this longstanding
problem and still it remains widely open. In fact, it was known that only five vertextransitive graphs exist without a Hamiltonian cycle which do not belong to Cayley
graphs. A Cayley graph is the subclass of vertex-transitive graph, and in view of the
Lov´asz conjecture, the attention has focused more toward the Hamiltonicity of Cayley
graphs. This survey will describe the current status of the search for Hamiltonian cycles and paths in Cayley graphs and digraphs on different groups, and discuss the future
direction regarding famous conjecture.
