A polynomial-time nearly-optimal algorithm for an edge coloring problem in outerplanar graphs

Given a graph G, we study the problem of finding the minimum number of colors required for a proper edge coloring of G such that any pair of vertices at distance 2 have distinct sets consisting of colors of their incident edges. This minimum number is called the 2-distance vertex-distinguishing index, denoted by $$\chi '_{d2}(G)$$χd2′(G). Using the breadth first search method, this paper provides a polynomial-time algorithm producing nearly-optimal solution in outerplanar graphs. More precisely, if G is an outerplanar graph with maximum degree $$\varDelta $$Δ, then the produced solution uses colors at most $$\varDelta +8$$Δ+8. Since $$\chi '_{d2}(G)\ge \varDelta $$χd2′(G)≥Δ for any graph G, our solution is within eight colors from optimal.