Abstract
A tree [Formula: see text] in a total colored graph [Formula: see text] is called a total-monochromatic tree if all the internal vertices and edges of [Formula: see text] have the same color. For [Formula: see text], a total-monochromatic [Formula: see text]-tree in [Formula: see text] is a total-monochromatic tree of [Formula: see text] containing the vertices of [Formula: see text]. For a connected graph [Formula: see text] and a given integer [Formula: see text] with [Formula: see text], the [Formula: see text]-monochromatic total-index [Formula: see text] of [Formula: see text] is the maximum number of colors needed such that for each subset [Formula: see text] of [Formula: see text] vertices, there exists a total-monochromatic [Formula: see text]-tree. In this paper, we first show some basic results about [Formula: see text]. Next, we determine all connected graphs [Formula: see text] for which [Formula: see text] is [Formula: see text] and [Formula: see text]. Moreover, we investigate the [Formula: see text]-monochromatic total-index of a graph [Formula: see text] according to some constraints of [Formula: see text]. Finally, we determine the [Formula: see text]-monochromatic total-index of a graph [Formula: see text] with order [Formula: see text] and [Formula: see text] for [Formula: see text].
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