Geodesic
D-resolvability of vertices in planar graphs
Journal of Graph Algorithms and Applications, Tome 21 (2017) no. 4, pp. 649-661.

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It is well known that a vertex of degree 5 in a planar graph is not $D$-reducible. However, it remains an open question whether all vertices in such a graph are $D$-resolvable. If so, it would be a stronger result than mere 4-colorability. To investigate the question, we designed and implemented two algorithms, one to 4-color a planar graph and the other to generate internally 6-connected triangulations randomly. The coloring algorithm greedily obtains a $k$-coloring and then, if $k > 4$ (highly likely), transforms the $k$-coloring into a 4-coloring solely by means of Kempe exchanges (generally possible only if all vertices are $D$-resolvable). We used the random generator to test the coloring algorithm systematically in over 200,000 trials that included many tens of thousands of non-isomorphic graphs. We also tested the algorithm on the graphs of Errera, Kittell, Heawood and others. We encountered no failures in any of the tests. 
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     author = {James Tilley},
     title = {D-resolvability of vertices in planar graphs},
     journal = {Journal of Graph Algorithms and Applications},
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     doi = {10.7155/jgaa.00433},
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     url = {http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00433/}
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James Tilley. D-resolvability of vertices in planar graphs. Journal of Graph Algorithms and Applications, Tome 21 (2017) no. 4, pp. 649-661. doi : 10.7155/jgaa.00433. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00433/

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