On the Number of Hamilton Cycles in Sparse Random Graphs

Random Structures and Algorithms, 2007

Suppose that a random graph begins with n isolated vertices and evolves by edges being added at random, conditional upon all vertex degrees being at most 2. The final graph is usually 2-regular, but is not uniformly distributed. Some properties of this final graph are already known, but the asymptotic probability of being a Hamilton cycle was not known. We answer this question along with some related questions about cycles arising in the process.

Discrete Applied Mathematics, 2018

A graph G is locally connected if for every v ∈ V (G) the open neighbourhood N (v) of v is nonempty and induces a connected graph in G. We characterize locally connected graphs of order n with less than 2n edges and show that for any natural number k the Hamilton Cycle Problem for locally connected graphs of order n with m edges is polynomially solvable if m ≤ 2n + k log 2 n, but NP-complete if m = 2n + n 1/k .

European Journal of Combinatorics, 2006

For a graph G the random n-lift of G is obtained by replacing each of its vertices by a set of n vertices, and joining a pair of sets by a random matching whenever the corresponding vertices of G are adjacent. We show that asymptotically almost surely the random lift of a graph G is hamiltonian, provided G has the minimum degree at least 5 and contains two disjoint Hamiltonian cycles whose union is not a bipartite graph.

The Electronic Journal of Combinatorics, 2020

We consider Hamilton cycles in the random digraph $D_{n,m}$ where the orientation of edges follows a pattern other than the trivial orientation in which the edges are oriented in the same direction as we traverse the cycle. We show that if the orientation forms a periodic pattern, other than the trivial pattern, then approximately half the usual $n\log n$ edges are needed to guarantee the existence of such Hamilton cycles a.a.s.

Random Structures & Algorithms, 2016

We introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of in order to typically find a subgraph possessing a given target property. We show that if , then one can find a Hamilton cycle with high probability after exposing edges. Our result is tight in both p and the number of exposed edges. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 635–668, 2016

SIAM Journal on Discrete Mathematics, 2007

We show for an arbitrary ℓp norm that the property that a random geometric graph G(n, r) contains a Hamiltonian cycle exhibits a sharp threshold at r = r(n) = log n αp n , where αp is the area of the unit disk in the ℓp norm. The proof is constructive and yields a linear time algorithm for finding a Hamiltonian cycle of G(n, r) a.a.s., provided r = r(n) ≥ log n (αp−ǫ)n for some fixed ǫ > 0.

The Electronic Journal of Combinatorics, 2004

Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d − 1) 2g−1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresponding result is given for random regular bipartite graphs.

Discrete Applied Mathematics, 1997

We prove that if a graph G on n > 32 vertices is hamiltonian and has two nonadjacent vertices u and u with d(u) + d(u) 3 n + z where z = 0 if n is odd and z = 1 if n is even, then G contains all cycles of length m where 3 < m < 1/5(n + 13).

The girth of a graph with a Hamiltonian cycle and t chords is investigated. In particular, for any integer t&gt;0 let g(t) denote the smallest number such that any Hamiltonian graph G with n vertices and n+t edges has girth at most g(t)n+c, where c is a constant independent of n. It is shown that there exist constants c 1 and c 2 such that (c 1 (logt))/t≤g(t)≤(c 2 (logt)/t). For small values of t (1≤t≤8), g(t) is determined precisely.

For a given graph G of minimum degree at least k, let G p denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ≥ log k+log log k+ω k (1) k , where ω k (1) is any function tending to infinity with k, then G p asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

Journal of Combinatorics, 2016

A k-uniform hypergraph H contains a Hamilton ℓ-cycle, if there is a cyclic ordering of the vertices of H such that the edges of the cycle are segments of length k in this ordering and any two consecutive edges f i , f i+1 share exactly ℓ vertices. We consider problems about packing and counting Hamilton ℓ-cycles in hypergraphs of large minimum degree. Given a hypergraph H, for a d-subset A ⊆ V (H), we denote by d H (A) the number of distinct edges f ∈ E(H) for which A ⊆ f , and set δ d (H) to be the minimum d H (A) over all A ⊆ V (H) of size d. We show that if a k-uniform hypergraph on n vertices H satisfies δ k−1 (H) ≥ αn for some α > 1/2, then for every ℓ < k/2 H contains (1 − o(1)) n • n! • α ℓ!(k−2ℓ)! n k−ℓ Hamilton ℓ-cycles. The exponent above is easily seen to be optimal. In addition, we show that if δ k−1 (H) ≥ αn for α > 1/2, then H contains f (α)n edge-disjoint Hamilton ℓ-cycles for an explicit function f (α) > 0. For the case where every (k − 1)-tuple X ⊂ V (H) satisfies d H (X) ∈ (α ± o(1))n, we show that H contains edge-disjoint Haimlton ℓ-cycles which cover all but o (|E(H)|) edges of H. As a tool we prove the following result which might be of independent interest: For a bipartite graph G with both parts of size n, with minimum degree at least δn, where δ > 1/2, and for p = ω(log n/n) the following holds. If G contains an r-factor for r = Θ(n), then by retaining edges of G with probability p independently at random, w.h.p the resulting graph contains a (1 − o(1))rp-factor.

Random Structures & Algorithms, 2014

The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum degree ∆ satisfies some basic expansion properties and contains a family of (1 − o(1))∆/2 edge disjoint Hamilton cycles, then there also exists a covering of its edges by (1 + o(1))∆/2 Hamilton cycles. This implies that for every α > 0 and every p ≥ n α−1 there exists a covering of all edges of G(n, p) by (1 + o(1))np/2 Hamilton cycles asymptotically almost surely, which is nearly optimal.

The Electronic Journal of Combinatorics, 1995

Let the edges of a graph $G$ be coloured so that no colour is used more than $k$ times. We refer to this as a $k$-bounded colouring. We say that a subset of the edges of $G$ is multicoloured if each edge is of a different colour. We say that the colouring is $\cal H$-good, if a multicoloured Hamilton cycle exists i.e., one with a multicoloured edge-set. Let ${\cal AR}_k$ = $\{G :$ every $k$-bounded colouring of $G$ is $\cal H$-good$\}$. We establish the threshold for the random graph $G_{n,m}$ to be in ${\cal AR}_k$.

Random Structures &amp; Algorithms, 2021

We investigate the emergence of subgraphs in sparse pseudo‐random k‐uniform hypergraphs, using the following comparatively weak notion of pseudo‐randomness. A k‐uniform hypergraph H on n vertices is called ‐pseudo‐random if for all (not necessarily disjoint) vertex subsets with we have urn:x-wiley:rsa:media:rsa21052:rsa21052-math-0004For any linear k‐uniform F, we provide a bound on in terms of and F, such that (under natural divisibility assumptions on n) any k‐uniform ‐pseudo‐random n‐vertex hypergraph H with a mild minimum vertex degree condition contains an F‐factor. The approach also enables us to establish the existence of loose Hamilton cycles in sufficiently pseudo‐random hypergraphs and, along the way, we also derive conditions which guarantee the appearance of any fixed sized subgraph. All results imply corresponding bounds for stronger notions of hypergraph pseudo‐randomness such as jumbledness or large spectral gap. As a consequence, ‐pseudo‐random k‐graphs as above cont...

Combinatorica, 2008

Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of |C(G)| over all graphs G of average degree d and girth g. Erdős conjectured that |C(G)| = Ω d (g-1)/2 for all such graphs, and we prove this conjecture. We also show that this is a lower bound for the number of odd cycle lengths in a graph of chromatic number d and girth g. Further results are obtained for the number of cycle lengths in H-free graphs of average degree d. In the second part of the paper, motivated by the conjecture of Erdős and Gyárfás (see also Erdős [9]) that every graph of minimum degree at least three contains a cycle of length a power of two, we prove a general theorem which gives an upper bound on the average degree of an n-vertex graph with no cycle of even length in a prescribed infinite sequence of integers. For many sequences, including the powers of two, our theorem gives the upper bound e O(log * n) on the average degree of graph of order n with no cycle of length in the sequence, where log * n is the number of times the binary logarithm must be applied to n to get a number which is at most one.