Given a modularity class k mod d, we present improved bounds on f(k, d), the smallest integer such that every graph with δ ≥ f( k, d) has a cycle whose length is equivalent to k mod d. Also, we characterize the endblocks of graphs with δ ≥ d and with no cycle whose length is equivalent to 2 mod d and prove a similar result for bipartite graphs. For each of the modularity classes 1 mod 3 and 2 mod 4, we characterize the endblocks of graphs with δ ≥ 3 and no cycle whose length is in the class. We show that every Hamiltonian graph with δ ≥ 3 has a cycle whose length is equivalent to 4 mod 5. Finally, we prove that any graph that is a member of one of the following classes of claw-free graphs has a cycle whose length is a power of two: graphs with δ ≥ 4 or δ ≥ 5; planar graphs with δ ≥ 3; and cubic graphs with δ ≥ 6.
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机译：给定模块化类 k mod d ，我们提出了最小的 f （ k，d ）的改进边界整数，以便每个具有δ≥ f （ k，d ）的图都有一个循环，该循环的长度等于 k mod d < /斜体>。此外，我们用δ≥ d 且无周期等于2 mod d 的图的端块来表征，并证明了二部图的相似结果。对于每个模块化类别1 mod 3和2 mod 4，我们用δ≥3且长度不属于该类别的图来表征图的端块。我们证明了每个δ≥3的哈密顿图都有一个长度等于4 mod 5的循环。最后，我们证明了属于以下无爪图类之一的任何图都有一个循环，其长度为2的幂：δ≥4或δ≥5的图； δ≥3的平面图；和δ≥6的立方图。
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