A famous conjecture of Pósa from 1962 asserts that every graph on n vertices and with minimum degree at least 2n/3 contains the square of a Hamilton cycle. The conjecture was proven for large graphs in 1996 by Komlós, Sárközy and Szemerédi [23]. In this paper we prove a degree sequence version of Pósa’s conjecture: Given any η > 0, every graph G of sufficiently large order n contains the square...

We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph Gn,p contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) lnn/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from Gn,p without destroy...

In this paper, we solve a problem by Glover and Punnen (1997) from the context of domination analysis, where the performance of a heuristic algorithm is rated by the number of solutions that are not better than the solution found by the algorithm, rather than by the relative performance compared to the optimal value. In particular, we show that for the Asymmetric Traveling Salesman Problem (ATS...

We show that the threshold for the random graph Gn,p to contain the square of a Hamilton cycle is p = 1 √ n . This improves the previous results of Kühn and Osthus and also Nenadov and Škorić.

Abstract. In an earlier paper the authors constructed a hamilton cycle embedding of Kn,n,n in a nonorientable surface for all n ≥ 1 and then used these embeddings to determine the genus of some large families of graphs. In this two-part series, we extend those results to orientable surfaces for all n 6= 2. In part I, we explore a connection between orthogonal latin squares and embeddings. A pro...

Let K (k) n be the complete k-uniform hypergraph, k ≥ 3, and let ` be an integer such that 1 ≤ ` ≤ k − 1 and k − ` divides n. An `-overlapping Hamilton cycle in K n is a spanning subhypergraph C of K (k) n with n/(k − `) edges and such that for some cyclic ordering of the vertices each edge of C consists of k consecutive vertices and every pair of consecutive edges in C intersects in precisely ...

A cyclic construction is presented for building embeddings of the complete tripartite graph Kn,n,n on a nonorientable surface such that the boundary of every face is a hamilton cycle. This construction works for several families of values of n, and we extend the result to all n with some methods of Bouchet and others. The nonorientable genus of Kt,n,n,n, for t ≥ 2n, is then determined using the...

For any two 2-regular spanning subgraphs G and H of the complete multipartite graph K, necessary and sufficient conditions are found for the existence of a 2-factorization of K in which (1) the first and second 2-factors are isomorphic to G and H respectively, and (2) each other 2-factor is a hamilton cycle, in the case where K has an odd number of vertices.