Ryjáček’s weak pancyclicity conjecture for locally connected graphs

Determine whether every locally connected graph is weakly pancyclic, as asserted by Ryjáček’s conjecture.

Background

The paper places locally nonforesty graphs within the broader study of graph properties inferred from local subgraphs. It notes that locally connected graphs are a related class and recalls the theorem of Oberly and Sumner that every connected, locally connected, claw-free graph is hamiltonian.

Against this background, the authors mention Ryjáček’s conjecture, which concerns a stronger cycle-structure property: weak pancyclicity, meaning that the graph contains cycles of every length between its girth and circumference. The conjecture is identified as unsolved and is not addressed by the paper, whose main results instead determine minimum edge counts for connected and k-connected locally nonforesty graphs.