Distance-2 Edge Coloring is NP-Complete

We prove that it is NP-complete to determine whether there exists a distance-2 edge coloring (strong edge coloring) with 5 colors of a bipartite 2-inductive graph with girth 6 and maximum degree 3. Let G be a simple, undirected graph. We say that two edges of G are within distance 2 of each other if either they are adjacent or there is some other edge that is adjacent to both of them. A distance-2-edge-coloring of G is an assignment of colors to edges so that any two edges within distance 2 of each other have distinct colors, or equivalently, a vertex-coloring of the square of the line graph of G. If the coloring uses only k colors, it is called a k-D2-edge-coloring, and the graph G is said to be k-D2-edge-colorable. Any k-D2-edge-colorable graph is also (k +1)-D2-edge-colorable. A distance-2-edge-coloring is also known as a strong edge coloring. Mahdian (2, 3) proved, via a reduction from Graph k-colorability, that it is NP-complete to determine, for every fixed g, whether a bipartite graph with girth g has a strong edge coloring with k colors for k � 4. We present a new proof that shows that the strong edge coloring problem is NP-complete for bipartite 2-inductive graphs of maximum degree 3. Definition 1 (c-inductive graphs). A graph G is said to be c-inductive if the vertices can be numbered so that at most c neighbors of any vertex v have higher numbers than v. Theorem 2. Determining whether a bipartite 2-inductive graph with girth 6 and maximum de- gree 3 is 5-D2-edge-colorable is NP-complete. Proof: The problem is clearly in NP since a coloring can be verified in polynomial time. To prove NP-hardness, we describe a reduction from Not-All-Equal-3SAT (1, Problem LO3).