论文标题
在两个连续的均匀长度上
On two cycles of consecutive even lengths
论文作者
论文摘要
邦迪（Bondy）和文斯（Vince）表明，每个图最低度至少三个图表包含两个长度的周期，差异为一两个。我们的证明主要基于结构分析，并且可能具有独立感兴趣的关键步骤表明，对于每个3个连接的图表，都有相同的结论，至少有6个顶点。这解决了一种猜想的特殊情况。定量结合是紧密的，还为长度两个模型四的循环提供了最佳的极端数。
Bondy and Vince showed that every graph with minimum degree at least three contains two cycles of lengths differing by one or two.We prove the following average degree counterpart that every $n$-vertex graph $G$ with at least $\frac52(n-1)$ edges, unless $4|(n-1)$ and every block of $G$ is a clique $K_5$, contains two cycles of consecutive even lengths. Our proof is mainly based on structural analysis, and a crucial step which may be of independent interest shows that the same conclusion holds for every 3-connected graph with at least 6 vertices. This solves a special case of a conjecture of Verstraëte. The quantitative bound is tight and also provides the optimal extremal number for cycles of length two modulo four.
