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Let \( G \) be an edge-colored connected graph. A path \( P \) is a proper path in \( G \) if no two adjacent edges of \( P \) are colored the same. If \( P \) is a proper \( u \) — \( v \) path of length \( d(u,v) \), then \( P \) is a proper \( u \) — \( v \) geodesic. An edge coloring \( c \) is a proper-path coloring of a connected graph \( G \) if every pair \( u,v \) of distinct vertices of \( G \) are connected by a proper \( u \) — \( v \) path in \( G \) and \( c \) is a strong proper coloring if every two vertices \( u \) and \( v \) are connected by a proper \( u \) — \( v \) geodesic in \( G \). The minimum number of colors used in a proper-path coloring and strong proper coloring of \( G \) are called the proper connection number \( \text{pc}(G) \) and strong proper connection number \( \text{spc}(G) \) of \( G \), respectively. These concepts are inspired by the concepts of rainbow coloring, rainbow connection number \( \text{rc}(G) \), strong rainbow coloring, and strong connection number \( \text