LONGEST CYCLES AND LONGEST CHORDLESS CYCLES IN $2$ -CONNECTED GRAPHS
YANAN HU, Chengli Li, FENG LIU · Bulletin of the Australian Mathematical Society · 2025
Abstract Thomassen’s chord conjecture from 1976 states that every longest cycle in a $3$ -connected graph has a chord. The circumference $c(G)$ and induced circumference $c'(G)$ of a graph G are the length of its longest cycles and the length of its longest chordless cycles, respectively. Harvey [‘A cycle of maximum order in a graph of high minimum degree has a chord’, Electron. J. Combin. 24 (4) (2017), Article no. 4.33, 8 pages] proposed the stronger conjecture: every $2$ -connected graph G with minimum degree at least $3$ has $c(G)\geq c'(G)+2$ . This conjecture implies Thomassen’s chord conjecture. We observe that wheels are the unique Hamiltonian graphs for which the circumference and the induced circumference differ by exactly one. Thus, we need only consider non-Hamiltonian graphs for Harvey’s conjecture. We propose a conjecture involving wheels that is equivalent to Harvey’s conjecture on non-Hamiltonian graphs. A graph is $\ell $ -holed if all its holes have length exactly $\ell $ . We prove that Harvey’s conjecture and hence also Thomassen’s conjecture hold for $\ell $ -holed graphs and graphs with a small induced circumference.