Compatible Spanning Circuits Visiting Each Vertex Exactly a Specified Number of Times in Graphs with Generalized Transition Systems

A transition in a graph refers to a pair of adjacent edges. A generalized transition system [Formula: see text] over a graph [Formula: see text], which can be regarded as a generalization of a partition system or an edge-coloring of [Formula: see text], defines a set of transitions over [Formula: see text]. A compatible spanning circuit in a graph [Formula: see text] with a generalized transition system [Formula: see text] refers to a spanning circuit in which no two consecutive edges form a transition defined by [Formula: see text]. In this paper, we present sufficient conditions for the existence of compatible spanning circuits that visit each vertex exactly [Formula: see text] times in some specific graphs on [Formula: see text] vertices with generalized transition systems, where [Formula: see text] denotes a function of a positive integer [Formula: see text], for every feasible [Formula: see text]. Moreover, as corollaries, we also obtain analogous conclusions for the above mentioned graphs that are assigned partition systems and edge-colorings, respectively.