Abstract
A graph G is uniquely k-colorable if the chromatic number of G is k and G has only one k-coloring up to permutation of the colors. A uniquely k-colorable graph G is edge-critical if G−e is not a uniquely k-colorable graph for any edge e∈E(G). Mel’nikov and Steinberg (Mel’nikov and Steinberg, 1977) asked to find an exact upper bound for the number of edges in an edge-critical uniquely 3-colorable planar graph with n vertices. In this paper, we give some properties of edge-critical uniquely 3-colorable planar graphs and prove that if G is such a graph with n(≥6) vertices, then |E(G)|≤52n−6, which improves the upper bound 83n−173 given by Matsumoto (Matsumoto, 2013). Furthermore, we find some edge-critical uniquely 3-colorable planar graphs with n(=10,12,14) vertices and 52n−7 edges.
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