A RELATIONSHIP BETWEEN THOMASSEN'S CONJECTURE AND BONDY'S CONJECTURE

In 1986, Thomassen posed the following conjecture: every 4-connected line graph has a Hamiltonian cycle. As a possible approach to the conjecture, many researchers have considered statements that are equivalent or related to it. One of them is the conjecture by Bondy: there exists a constant c(0) with 0 < c(0) <= 1 such that every cyclically 4-edge-connected cubic graph H has a cycle of length at least c(0)\V (H)\. It is known that Thomassen's conjecture implies Bondy's conjecture, but nothing about the converse has been shown. In this paper, we show that Bondy's conjecture implies a slightly weaker version of Thomassen's conjecture: every 4-connected line graph with minimum degree at least 5 has a Hamiltonian cycle.