Contractible edges in longest cycles

Abstract

AbstractThis paper investigates the number of contractible edges in a longest cycle of a ‐connected graph that is triangle‐free or has minimum degree at least . We prove that, except for two graphs, contains at least contractible edges. For triangle‐free 3‐connected graphs, we show that contains at least contractible edges, and characterize all graphs having a longest cycle containing exactly six/seven contractible edges. Both results are tight. Lastly, we prove that every longest cycle of a 3‐connected graph of girth at least 5 contains at least contractible edges.