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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.
Guo et al. (Fri,) studied this question.