Strong Edge Coloring Of Planar Graphs

A strong k-edge-coloring of a graph G is a mapping cp:E(G)→{1,2,...,k},such that φp(e)≠φ(e’)for every pair of distinct edges at distance at most two.The strong chromatical index of a graph G is the least integer k such that G has a strong-k-edge-coloring,denoted by Xs’(G).In this paper,we prove this result:for any subcubic planar graph with g(G)≥6,7--cycles are not adjacent to 9--cycles and 8-cycles is not adjacent to 8-cycles,then Xs’(G)≤8.To prove this theorems,All graphs mentioned in this paper are finite simple planar graphs.Let G be a counterexample such that |V(G)| + |E(G)| is minimized.If we can prove that the smallest counterexample does not exist,then our theorem can prove it.Prove that the theorem is mainly divided into two parts:The first part is to determine the structure of graph G by proving that the structure of graph G does not exist.The second part define an initial weight function μ(v)= 2d(v)-6 for each v ∈V(G),and μ(f)= d(f)-6 for each f∈F(G).From the Euler’s formula,we obtain ∑ μ(x)=-12.We will design several discharging rules and reassign weight without change in the sum of weight.Then we arrive at a new weight functionμ*(x)and show that for each x ∈ V(G)∪F(G),μ*(x)≥0.This leads to the following contradiction:(?)The contradiction shows the nonexistence of G,which completes the proof of Theo-rem.