Geodesic
On q-Power Cycles in Cubic Graphs
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 1, pp. 211-220.

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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.
Keywords: cubic graphs, q -power cycles, Erdős-Gyárfás conjecture 
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Bensmail, Julien. On q-Power Cycles in Cubic Graphs. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 1, pp. 211-220. http://geodesic.mathdoc.fr/item/DMGT_2017_37_1_a14/

[1] P. Erdős, Some old and new problems in various branches of combinatorics, Discrete Math. 165/166 (1997) 227–231. doi:10.1016/S0012-365X(96)00173-2

[2] D. Daniel and S.E. Shauger, A result on the Erdős-Gyárfás conjecture in planar graphs, Congr. Numer. 153 (2001) 129–139.

[3] C.C. Heckman and R. Krakovski, Erdős-Gyárfás conjecture for cubic planar graphs, Electron. J. Combin. 20 (2013) #P7.

[4] K. Markström, Extremal graphs for some problems on cycles in graphs, Congr. Numer. 171 (2004) 179–192.

[5] P.S. Nowbandegani, H. Esfandiari, M.H.S. Haghighi and K. Bibak, On the Erdős-Gyárfás conjecture in claw-free graphs, Discuss. Math. Graph Theory 34 (2014) 635–640. doi:10.7151/dmgt.1732

[6] S.E. Shauger, Results on the Erdős-Gyárfás conjecture in K1,m-free graphs, Congr. Numer. 134 (1998) 61–65.

[7] D. West, Erdős-Gyárfás conjecture on 2 -power cycle lengths, Open Problems—Graph Theory and Combinatorics. http://www.math.illinois.edu/~dwest/openp/2powcyc.html