Legally $$(\varDelta +2)$$ ( Δ + 2 ) -Coloring Bipartite Outerplanar Graphs in Cubic Time

Abstract

The 2-distance vertex-distinguishing index $$\chi '_{d2}(G)$$ of a graph G is the least number of colors required for a proper edge coloring of G such that any pair of vertices at distance 2 have distinct sets of colors on their incident edges. Let G be a bipartite outerplanar graph of order n with maximum degree $$\varDelta $$ . We give an algorithm of time complexity $$O(n^3)$$ to show that $$\chi '_{d2}(G) \le \varDelta +2$$ .