We show that in any graph G on n vertices with d(x) + d(y) ≥ n for any two nonadjacent vertices x and y, we can fix the order of k vertices on a given cycle and find a hamiltonian cycle encountering these vertices in the same order, as long as k < n/12 and G is d(k + 1)/2e-connected. Further we show that every b3k/2cconnected graph on n vertices with d(x) + d(y) ≥ n for any two nonadjacent vert...

The best worst case guarantee algorithm to see if a graph has a Hamiltonian cycle, a closed tourvisiting every vertex exactly once, for a long time was based on dynamic programming over allthe vertex subsets of the graph. In this talk we will show some algebraic techniques that can beused to see if a graph has a Hamiltonian cycle much faster. These techniques utilize sums over<l...

The star graph interconnection network has been recognized as an attractive alternative to the hypercube network. Previously, the star graph has been shown to contain a Hamiltonian cycle. In this paper, we consider an injured star graph with some faulty links and nodes. We show that even with fe £ n 3 faulty links, a Hamiltonian cycle still can be found in an n-star, and that with fv £ n 3 faul...

Let G be a graph of order n and 3 ≤ t ≤ n4 be an integer. Recently, Kaneko and Yoshimoto provided a sharp δ(G) condition such that for any set X of t vertices, G contains a hamiltonian cycle H so that the distance along H between any two vertices of X is at least n/2t. In this paper, minimum degree and connectivity conditions are determined such that for any graph G of sufficiently large order ...

This paper proposes three eecient parallel algorithms for computing the range-join of two relations on two-dimensional n m mesh-connected computers, where n and m are the numbers of the rows and columns respectively. After sorting all subsets of both relations, all proposed algorithms permute all sorted subsets of one relation to each processor in the computers, where they are joined with the s...

Problem statement: Modified Chordal Rings Degree Four, called CHRm4 is the first modified structure of chordal rings. This CHRm4 is an undirected circulant graph and is a double loop graph. Approach: This study presented the main properties of CHRm4. There are connectivity, Hamiltonian cycle and asymmetric. Results: Several definitions, postulates, corollary, theorems and lemmas were constructe...

We investigate hamiltonian properties of Pm × Cn, m ≥ 2 and even n ≥ 4, which is bipartite, in the presence of faulty vertices and/or edges. We show that Pm×Cn with n even is strongly hamiltonianlaceable if the number of faulty elements is one or less. When the number of faulty elements is two, it has a fault-free cycle of length at least mn−2 unless both faulty elements are contained in the sa...

We undertake a study on computing Hamiltonian alternating cycles and paths on bicolored point sets. This has been an intensively studied problem, not always with a solution, when the paths and cycles are also required to be plane. In this paper, we relax the constraint on the cycles and paths from being plane to being 1-plane, and deal with the same type of questions as those for the plane case...

It was proved by Glover and Marušič (J. Eur. Math. Soc. 9:775–787, 2007), that cubic Cayley graphs arising from groups G = 〈a, x | a2 = x = (ax)3 = 1, . . .〉 having a (2, s,3)-presentation, that is, from groups generated by an involution a and an element x of order s such that their product ax has order 3, have a Hamiltonian cycle when |G| (and thus also s) is congruent to 2 modulo 4, and have ...

The crossed cube is one of the most notable variations of hypercube, but some properties of the former are superior to those of the latter. For example, the diameter of the crossed cube is almost the half of that of the hypercube. In this paper, we focus on the problem embedding a Hamiltonian cycle through an arbitrary given edge in the crossed cube. We give necessary and sufficient condition f...