Long Cycles in 2-Connected Triangle-Free Graphs.
Douglas C. Bauer, Nathan Kahl, Linda E. McGuire, Edward F. Schmeichel · Ars Combinatoria · 2008
Dirac showed that a 2–connected graph of order n with minimum degree δ has circumference at least min{2δ, n}. We prove that a 2– connected, triangle-free graph G of order n with minimum degree δ either has circumference at least min{4δ−4, n}, or every longest cycle in G is dominating. This result is best possible in the sense that there exist bipartite graphs with minimum degree δ whose longest cycles have length 4δ − 4, and are not dominating.