A red-blue coloring of a graph \( G \) is an edge coloring of \( G \) in which every edge of \( G \) is colored red or blue. Let \( F \) be a connected graph of size 2 or more with a red-blue coloring, at least one edge of each color, where some blue edge of \( F \) is designated as the root of \( F \). Such an edge-colored graph \( F \) is called a color frame. An \( F \)-coloring of a graph \( G \) is a red-blue coloring of \( G \) in which every blue edge of \( G \) is the root edge of a copy of \( F \) in \( G \). The \( F \)-chromatic index \( \chi’_F(G) \) of \( G \) is the minimum number of red edges in an \( F \)-coloring of \( G \). It has been shown that these concepts generalize both edge domination and matchings in graphs. In this paper, we consider the two color frames \( Y_1 \) and \( Y_2 \) that result from the claw \( K_{1,3} \), where \( Y_1 \) has exactly one red edge and \( Y_2 \) has exactly two red edges. An edge \( e \) in a graph \( G \) is a non-claw edge if \( e \) belongs to no claw in \( G \). It is shown that if \( G \) is a connected graph containing \( \ell \) non-claw edges, then \( \chi’_{Y_1}(G) \leq \chi’_{Y_2}(G) \leq 3\chi’_{Y_1}(G) – 2\ell \) and \( \chi’_{Y_1}(G) = \chi’_{Y_2}(G) \) if and only if \( G \) is a path or cycle. Furthermore, a pair \( a, b \) of positive integers can be realized as the \( Y_1 \)-chromatic index and \( Y_2 \)-chromatic index for some connected graph of order at least 4 if and only if \( a \leq b \leq 3a \) and \( b \geq 2 \).