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A \emph{red-blue coloring} of a graph \( G \) is an edge coloring of \( G \) in which every edge is colored red or blue. For a connected graph \( H \) of size at least 2, a \emph{color frame} \( F \) of \( H \) is obtained from a red-blue coloring of \( H \) having at least one edge of each color and in which a blue edge is designated as the root edge.
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 \( G \), and the \( F \)-\emph{chromatic index} of \( G \) is the minimum number of red edges in an \( F \)-coloring of \( G \). An \( F \)-coloring of \( G \) is \emph{minimal} if whenever any red edge of \( G \) is changed to blue, then the resulting red-blue coloring of \( G \) is not an \( F \)-coloring of \( G \). The maximum number of red edges in a minimal \( F \)-coloring of \( G \) is the \emph{upper \( F \)-chromatic index} of \( G \).
In this paper, we investigate \( F \)-colorings and \( F \)-chromatic indexes of graphs for all color frames \( F \) of paths of orders 3 and 4.