(1, 0)-Relaxed strong edge list coloring of planar graphs with girth 6

Let G be a graph. An (s, t)-relaxed strong edge k-coloring is a mapping pi : E(G) -> {1, 2, ..., k} such that for any edge e, there are at most s edges adjacent to e and t edges which are distance two apart from e assigned the same color as e. The (s, t)-relaxed strong chromatic index, denoted by chi((s,t))'(G), is the minimum number k of an (s, t)-relaxed strong k-edge-coloring admitted by G. G is called (s, t)-relaxed strong edge L-colorable if for a given list assignment L = {L(e) vertical bar e is an element of E(G)}, there exists an (s, t)-relaxed strong edge coloring pi of G such that pi(e) is an element of L(e) for all e is an element of E(G). If G is (s, t)-relaxed strong edge L-colorable for any list assignment with vertical bar L(e)vertical bar = k for all e is an element of E(G), then G is said to be (s, t)-relaxed strong edge k-choosable. The (s, t)-relaxed strong list chromatic index, denoted by ch((s,t))'(G), is defined to be the smallest integer k such that G is (s, t)-relaxed strong edge k-choosable. In this paper, we prove that every planar graph G with girth 6 satisfies that ch((1,0))'(G) <= 3 Delta(G) - 1. This strengthens a result which says that every planar graph G with girth 7 and Delta (G) >= 4 satisfies that chi((1,0))'(G) <= 3 Delta(G) - 1.