Contractible edges in longest cycles

This paper investigates the number of contractible edges in a longest cycle C $C$ of a k $k$-connected graph (k >= 3) $(k\ge 3)$ that is triangle-free or has minimum degree at least 32k-1 $\frac{3}{2}k-1$. We prove that, except for two graphs, C $C$ contains at least min{|E(C)|,6} $\min \{|E(C)|,6\}$ contractible edges. For triangle-free 3-connected graphs, we show that C $C$ contains at least min{|E(C)|,7} $\min \{|E(C)|,7\}$ 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 C $C$ of a 3-connected graph of girth at least 5 contains at least |E(C)|12 $\frac{|E(C)|}{12}$ contractible edges.