The adjacent vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that the edge coloring set on any pair of adjacent vertices is distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of G is denoted by \(\chi _{a}'(G)\). It is observed that \(\chi _a'(G)\ge \Delta (G)+1\) when G contains two adjacent vertices of degree \(\Delta (G)\). In this paper, we prove that if G is a planar graph without 4-cycles, then \(\chi _a'(G)\le \max \{9,\Delta (G)+1\}\).

中文翻译：

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不带4圈的平面图的相邻顶点区分边缘着色

相邻顶点区别边的曲线图的着色ģ是一个适当的边缘着色的ģ使得边缘上的任何一对相邻顶点着色组显着。G的相邻顶点区分边缘着色所需的最小颜色数由\（\ chi _ {a}'（G）\）表示。观察到，当G包含度数为\（\ Delta（G）\）的两个相邻顶点时，\（\ chi _a'（G）\ ge \ Delta（G）+1 \）。在本文中，我们证明如果G是没有4个循环的平面图，则\（\ chi _a'（G）\ le \ max \ {9，\ Delta（G）+1 \} \）。