Abstract
A path in an edge-colored graph is proper if no two adjacent edges of the path receive the same color. For a connected graph G, the strong proper connection number (SPC number) of G, denoted spc(G), is the minimum number of colors needed to color the edges of G so that every pair of distinct vertices of G is connected by at least one proper geodesic in G. A connected graph G is k-SPC if spc(G)≤k. It is implied by the NP-completeness of 3-edge-coloring problem of cubic graphs that the problem to recognize 3-SPC cubic graphs is NP-complete. Then we present in this paper a complete characterization for 2-SPC cubic graphs based on establishing some nice structures of forced branches used in our research. This leads to a linear-time algorithm to recognize 2-SPC cubic graphs. As consequences, for cubic claw-free graphs and cubic bipartite graphs, their SPC numbers can be determined in linear time.
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