In the context of a conjecture of Erdős and Gyárfás, we consider, for any q ≥ 2, the existence of q-power cycles (i.e., with length a power of q) in cubic graphs. We exhibit constructions showing that, for every q ≥ 3, there exist arbitrarily large cubic graphs with no q-power cycles. Concerning the remaining case q = 2 (which corresponds to the conjecture of Erdős and Gyárfás), we show that there exist arbitrarily large cubic graphs whose all 2-power cycles have length 4 only, or 8 only.
@article{bwmeta1.element.doi-10_7151_dmgt_1926,
author = {Julien Bensmail},
title = {Onq-Power Cycles in Cubic Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {37},
year = {2017},
pages = {211-220},
zbl = {06676796},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1926}
}
Julien Bensmail. Onq-Power Cycles in Cubic Graphs. Discussiones Mathematicae Graph Theory, Tome 37 (2017) pp. 211-220. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1926/