4 research outputs found
Hamiltonicity in connected regular graphs
In 1980, Jackson proved that every 2-connected -regular graph with at most
vertices is Hamiltonian. This result has been extended in several papers.
In this note, we determine the minimum number of vertices in a connected
-regular graph that is not Hamiltonian, and we also solve the analogous
problem for Hamiltonian paths. Further, we characterize the smallest connected
-regular graphs without a Hamiltonian cycle.Comment: 5 page
Cycles containing all vertices of maximum degree
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
Problems in combinatorics: Paths in graphs, partial orders of fixed width.
This thesis contains results in two areas, that is, graph theory and partial orders. (1) We consider graphs G with a specified subset W of vertices of large degree. We look for paths in G containing many vertices of W. The main results of the thesis are as follows. For G a graph on n vertices, and W of size w and minimum degree d, we show that there is always a path through at least vertices of W. We also prove some results for graphs in which only the degree sums of sets of independent vertices in W are known. (2) Let P = (X, ) be a poset on a set {lcub}1, 2,..., N{rcub}. Suppose X1 and X2 are a pair of disjoint chains in P whose union is X. Then P is a partial order of width two. A labelled poset is a partial order on a set {lcub}1, 2,..., N{rcub}. Suppose we have two labelled posets, P1 and P2, that are isomorphic. That is, there is a bijection between P1 and P2 which preserves all the order relations. Each isomorphism class of labelled posets corresponds to an unlabelled poset. (Abstract shortened by UMI.)