A new lower bound on the density of vertex identifying codes for the infinite hexagonal grid.

Cranston, Daniel W.; Yu, Gexin

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5-Stars of Low Weight in Normal Plane Maps with Minimum Degree 5

Oleg V. Borodin, Anna O. Ivanova, Tommy R. Jensen (2014)

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It is known that there are normal plane maps M5 with minimum degree 5 such that the minimum degree-sum w(S5) of 5-stars at 5-vertices is arbitrarily large. In 1940, Lebesgue showed that if an M5 has no 4-stars of cyclic type (5, 6, 6, 5) centered at 5-vertices, then w(S5) ≤ 68. We improve this bound of 68 to 55 and give a construction of a (5, 6, 6, 5)-free M5 with w(S5) = 48