The k-Total-Proper Index of Graphs

Given a set [Formula: see text] with [Formula: see text], a tree [Formula: see text] is considered as a total proper [Formula: see text]-tree or a total proper tree connecting [Formula: see text] if any two adjacent or incident elements of edges and [Formula: see text] of [Formula: see text] differ in color. Let [Formula: see text] be a connected graph of order [Formula: see text], and [Formula: see text] be an integer with [Formula: see text]. A total-colored graph is total proper[Formula: see text]-tree connected if for every set [Formula: see text] of [Formula: see text] vertices, there exists a total proper [Formula: see text]-tree in [Formula: see text]. The [Formula: see text]-total-proper index of [Formula: see text], denoted by [Formula: see text], is the minimum number of colors required to make [Formula: see text] total proper [Formula: see text]-tree connected. In this paper, we first investigate the [Formula: see text]-total-proper index of some special graphs. Moreover, we characterize the graphs with [Formula: see text]-total-proper index [Formula: see text] and [Formula: see text], respectively.