On 2-power unicyclic cubic graphs

Shariefuddin Pirzada (Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India)
Mushtaq Shah (Department of Mathematics, AAAM Degree College, Bemina, Srinagar, India)
Edy Tri Baskoro (Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 10, No 1 (2022): Electronic Journal of Graph Theory and Applications

In a graph, a cycle whose length is a power of two (that is, 2k) is called a 2-power cycle. In this paper, we show that the existence of an infinite family of cubic graphs which contain only one cycle whose length is a power of 2. Such graphs are called as 2-power unicyclic cubic graphs. Further we observe that the only 2-power cycle in a cubic graph cannot be removed implying that there does not exist a counter example for Erdos-Gyárfás conjecture.