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For a graph G = (V , E) , an edge connectivity λ is the minimum number of edges whose removal disconnects G. In this paper, we define property restricted edge connectivity. Then we appraise its particular case, more precisely k-restricted edge connectivity. Some of the basic properties of k-restricted edge connectivity are considered. We also discuss how k-restricted edge connectivity associates with graph coloring. In particular, we try to study how k-restricted edge connectivity relates with Tait's coloring theorem and thereby the famous Four-color conjecture. [ABSTRACT FROM AUTHOR]