Abstract
We prove that every k-connected graph (k ≥ 5) has an even cycle C such that G-V (C) is still (k-4)-connected. The connectivity bound is best possible.
In addition, we prove that every k-connected triangle-free graph (k ≥ 5) has an even cycle C such that G-V (C) is still (k-3)-connected. The same conclusion also holds for any k-connected graph (k ≥ 5) that does not contain a K −4 , i.e., K 4 minus one edge.
The first theorem is an analogue of the well-known result (without the parity condition) by Thomassen [9]. It is also a counterpart of the conjecture made by Thomassen [10] which says that there exists a function f(k) such that every f(k)-connected non-bipartite graph has an odd cycle C such that G-V (C) is still k-connected.
The second theorem is an analogue of the results (without the parity condition) by Egawa [2] and Kawarabayashi [3], respectively.
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This work is supported by the JSPS Research Fellowships for Young Scientists.
Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, by C&C Foundation, by Inamori Foundation and by Kayamori Foundation.
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Fujita, S., Kawarabayashi, Ki. Non-separating even cycles in highly connected graphs. Combinatorica 30, 565–580 (2010). https://doi.org/10.1007/s00493-010-2482-6
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DOI: https://doi.org/10.1007/s00493-010-2482-6