It is shown that for every > 0 and n > n0( ), any complete graph K on n vertices whose edges are colored so that no vertex is incident with more than (1 − 1 √ 2 − )n edges of the same color, contains a Hamilton cycle in which adjacent edges have distinct colors. Moreover, for every k between 3 and n any such K contains a cycle of length k in which adjacent edges have distinct colors.

It is known that Θ(log n) chords must be added to an n-cycle to produce a pancyclic graph; for vertex pancyclicity, where every vertex belongs to a cycle of every length, Θ(n) chords are required. A possibly ‘intermediate’ variation is the following: given k, 1 ≤ k ≤ n, how many chords must be added to ensure that there exist cycles of every possible length each of which passes exactly k chords...

Graph invariants provide a powerful analytical tool for investigation of abstract substructures of graphs. This paper is devoted to large cycle substructures, namely, Hamilton, longest and dominating cycles and some generalized cycles including Hamilton and dominating cycles as special cases. In this paper, we have collected 36 pure algebraic relations between basic initial graph invariants ens...

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have als...

Let G be a graph of order n and define NC(G) = min{l/V(u) U N(u)l Iuu $& E(G)}. A cycle C of G is called a dominating cycle or D-cycle if MG) MC) is an independent set. A D-path is defined analogously. The following result is proved: if G is 2-connected and contains a D-cycle, then G contains a D-cycle of length at least rnin(n, WCIG)} unless G is the Petersen graph. By combining this result wi...

Let D(n, p) be the random directed graph on n vertices where each of the n(n− 1) possible arcs is present independently with probability p. It is known that if p ≥ (log n+ ω(1))/n then D(n, p) typically has a directed Hamilton cycle, and this is best possible. We show that under the same condition, the number of directed Hamilton cycles in D(n, p) is typically n!(p(1 + o(1))) . We also prove a ...

We say that a k-uniform hypergraph C is a Hamilton cycle of type `, for some 1 ≤ ` ≤ k, if there exists a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices and for every pair of consecutive edges Ei−1, Ei in C (in the natural ordering of the edges) we have |Ei−1 \ Ei| = `. We prove that for k/2 < ` ≤ k, with high probability almost all edges of the ran...

In this article, we discuss some aspects of the search for chessknight closed tours in a chessboard, as in [1], [g], [10], [14] and [16] and give an elementary application to the interaction between graph theory and group theory. These closed tours will be herewith denoted as chessknight Hamilton cycles. In particular, we prove the following: If n and r are integers>0withn4r> 2,thenthedifferenc...

For integers k ≥ 1 and n ≥ 2k + 1, the Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of {1, . . . , n} and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form K(2k + 1, k) are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every k ≥ 3, the odd graph K(2k + 1,...