The distance coloring problem of graph is a graph model for frequency allocation problem. The so-called frequency allocation problem refers to that different radio stations in a certain area need to use radio waves to send signals. In order to avoid interference， stations in close proximity need to use different channels， and when stations are particularly close， they need to be separated by at least two channels.

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-edge coloring means that the chromatic number difference of two edges with distance 1 is greater than or equal to 2， and the chromatic number of two edges with distance greater than 1 is different. In this paper， an

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-edge coloring algorithm is designed for random graphs. According to the experimental results， it is shown that the

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-edge coloring problem of random graphs in finite points can be solved through the algorithm. Meanwhile， the coloring properties of three kinds of unicyclic graphs are found here， and we define

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，

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and

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to characterize these three kinds of unicyclic graphs respectively. The related theorems and proofs are given as well.