Geodesic
Antipodal Edge-Colorings of Hypercubes
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 1, pp. 271-284.

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Two vertices of the k-dimensional hypercube Qk are antipodal if they differ in every coordinate. Edges uv and xy are antipodal if u is antipodal to x and v is antipodal to y. An antipodal edge-coloring of Qk is a 2- edge-coloring such that antipodal edges always have different colors. Norine conjectured that for k ≥ 2, in every antipodal edge-coloring of Qk some two antipodal vertices are connected by a monochromatic path. Feder and Subi proved this for k ≤ 5. We prove it for k ≤ 6.
Keywords: antipodal edge-coloring, hypercube, monochromatic geodesic 
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West, Douglas B.; Wise, Jennifer I. Antipodal Edge-Colorings of Hypercubes. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 1, pp. 271-284. http://geodesic.mathdoc.fr/item/DMGT_2019_39_1_a20/

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[2] T. Feder and C. Subi, On hypercube labellings and antipodal monochromatic paths, Discrete Appl. Math. 161 (2013) 1421-1426. doi: 10.1016/j.dam.2012.12.025

[3] K. Gandhi, Maximal monochromatic geodesics in an antipodal coloring of hypercube (2015), manuscript. http://math.mit.edu/research/highschool/primes/materials/2014/Gandhi.pdf

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