Cycles In Regular Graphs Research Articles

On the Minimum Number of Hamiltonian Cycles in Regular Graphs

ABSTRACTA graph construction that produces a k-regular graph on n vertices for any choice of k ⩾ 3 and n = m(k + 1) for integer m ⩾ 2 is described. The number of Hamiltonians cycles in such graphs can be explicitly determined as a function of n and k, and empirical evidence is provided that suggests that this function gives a tight upper bound on the minimum number of Hamiltonian cycles in k-regular graphs on n vertices for k ⩾ 5 and n ⩾ k + 3. An additional graph construction for 4-regular graphs is described for which the number of Hamiltonian cycles is superior to the above function in the case when k = 4 and n ⩾ 11.

Solution to a problem of Bollobás and Häggkvist on Hamilton cycles in regular graphs

We prove that, for large n, every 3-connected D-regular graph on n vertices with D≥n/4 is Hamiltonian. This is best possible and verifies the only remaining case of a conjecture posed independently by Bollobás and Häggkvist in the 1970s. The proof builds on a structural decomposition result proved recently by the same authors.

Longest cycles in regular graphs

The paper is concerned with the longest cycles in regular three- (or two-) connected graphs. In particular, the following results are proved: (i) every 3-connected k-regular graph on n vertices has a cycle of length at least min(3 k, n); (ii) every 2-connected k-regular graph on n vertices, where n < 3 k + 4, has a cycle of length at least min(3 k, n).

Edge-Disjoint Hamilton Cycles in Regular Graphs of Large Degree

Edge-Disjoint Hamilton Cycles in Regular Graphs of Large Degree

Hamilton cycles in regular graphs

Hamilton cycles in regular graphs

Hamiltonian cycles in regular graphs of moderate degree

In this paper we prove that if k is an integer no less than 3, and if G is a two-connected graph with 2n a vertices, a E {0, 1}, which is regular of degree n k, then G is Hamiltonian if a = 0 and n > k2 + k + 1 or if a = I and n > 2k 2 3k =, 3 . We use the notation and terminology of [1] . Gordon [4] has proved that there are only a small number of exceptional graphs with 2n vertices which are not Hamiltonian when all vertices have degree n 1 or more. The present authors proved [3] that if G is a two-connected graph with 2n vertices which is regular of degree n 2 and if n > 6, then G is Hamiltonian. We now partially extend that result to regular graphs of degree n k, k > 3 . Throughout this paper we suppose that G is a graph with 2n a vertices, with a c {0, 1), which is two connected and regular of degree n k, where k is an integer no less than three . Let P be a longest cycle in G, choose a direction around P, let R = V(G) V(P), and let r = I R 1 . For the lemmas, suppose r > 1 . By a theorem of Dirac [2], ((P) , 2n 2k. For v c R, let C„ be the set of vertices of P adjacent to v, let A,; be the set of vertices of P immediately preceding elements of C, in the ordering of P, and let B,, be the set of vertices of P immediately following elements of C,, . The first lemma is trivial .