Cycles containing all vertices of maximum degree


Broersma, H.J. and Heuvel, J. van den and Jung, H.A. and Veldman, H.J. (1993) Cycles containing all vertices of maximum degree. Journal of Graph Theory, 17 (3). pp. 373-385. ISSN 0364-9024

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Abstract:For a graph G and an integer k, denote by Vk the set {v ε V(G) | d(v) ≥ k}. Veldman proved that if G is a 2-connected graph of order n with n ≤ 3k - 2 and |Vk| ≤ k, then G has a cycle containing all vertices of Vk. It is shown that the upper bound k on |Vk| is close to best possible in general. For the special case k = δ(G), it is conjectured that the condition |Vk| ≤ k can be omitted. Using a variation of Woodall's Hopping Lemma, the conjecture is proved under the additional condition that n ≤ 2δ(G) + δ(G) + 1. This result is an almost-generalization of Jackson's Theorem that every 2-connected k-regular graph of order n with n ≤ 3k is hamiltonian. An alternative proof of an extension of Jackson's Theorem is also presented.
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Copyright:© 1993 Wiley InterScience
Electrical Engineering, Mathematics and Computer Science (EEMCS)
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