4 research outputs found
On q-power cycles in cubic graphs
International audienceIn the context of a conjecture of Erdős and Gyárfás, we consider, for any , the existence of q-power cycles (i.e. with length a power of q) in cubic graphs. We exhibit constructions showing that, for every , there exist arbitrarily large cubic graphs with no q-power cycles. Concerning the remaining case (which corresponds to the conjecture of Erdős and Gyárfás), we show that there exist arbitrarily large cubic graphs whose only 2-power cycles have length 4 only, or 8 only
Minimal unavoidable sets of cycles in plane graphs
A set of cycles is minimal unavoidable in a graph family if each graph contains a cycle from and, for each proper subset , there exists an infinite subfamily such that no graph from contains a cycle from . In this paper, we study minimal unavoidable sets of cycles in plane graphs of minimum degree at least 3 and present several graph constructions which forbid many cycle sets to be unavoidable. We also show the minimality of several small sets consisting of short cycles
On q-Power Cycles in Cubic Graphs
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