Given a family of systems, identifying stabilizing switching signals in terms of infinite walks constructed by concatenating cycles on the underlying directed graph of a switched system that satisfy certain conditions, is a well-known technique in the literature. This paper deals with a new {method to design} these cycles for stability of switched linear systems. We employ properties of the subsystem matrices and mild assumption on the admissible switches between the subsystems {for this purpose}. In contrast to prior works, {our construction of} stabilizing cycles does not involve design of Lyapunov-like functions and storage of sets of scalars in memory prior to the application of a cycle detection algorithm. As a result, {the} techniques {proposed in this paper} offer improved numerical tractability.