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-path coloring if every two vertices u and v are connected by a proper u − v geodesic in G. The minimum number of colors required for a proper-path coloring or strong proper-path coloring of G is called the proper connection number pc(G) or strong proper connection number spc(G) of G, respectively. If G is a nontrivial connected graph of size m, then pc(G) ≤ spc(G) ≤ m and pc(G) = m or spc(G) = m if and only if G is the star of size m. In this paper, we determine all connected graphs G of size m for which pc(G) or spc(G) is m − 1, m − 2 or m − 3.

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For a graph G of size m ≥ 1 and edge-induced subgraphs F and H of size k where 1 ≤ k ≤ m, the subgraph H is said to be obtained from the subgraph F by an edge jump if there exist four distinct vertices u, v, w and x such that uv ∈ E(F ), wx ∈ E(G) − E(F ), and H = F − uv + wx. The k-jump graph J k (G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G where two vertices F and H of J k (G) are adjacent if and only if H can be obtained from F by an edge jump.All connected graphs G for whose J3(G) is planar are determined.

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