We study Hamiltonicity in random subgraphs of the hypercube $\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\mathcal{Q}^n$ according to a uniformly chosen random ordering. Then, with high probability, as soon as the graph produced by this process has minimum degree $2k$, it contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. Secondly, we obtain a perturbation result: if $H\subseteq\mathcal{Q}^n$ satisfies $δ(H)\geqαn$ with $α>0$ fixed and we consider a random binomial subgraph $\mathcal{Q}^n_p$ of $\mathcal{Q}^n$ with $p\in(0,1]$ fixed, then with high probability $H\cup\mathcal{Q}^n_p$ contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. In particular, both results resolve a long standing conjecture, posed e.g. by Bollobás, that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals $1/2$. Our techniques also show that, with high probability, for all fixed $p\in(0,1]$ the graph $\mathcal{Q}^n_p$ contains an almost spanning cycle. Our methods involve branching processes, the Rödl nibble, and absorption.
In this paper we extend a result about a sufficient condition for Hamiltonicity for finite graphs by Broersma and Veldmann to locally finite graphs. In order to do this we use topological circles within the Freudenthal compactification of a locally finite graph as infinite cycles. The condition we focus on in this paper is in terms of forbidden induced subgraphs, namely being claw-free and a relaxation of being paw-free.
In a series of papers, of which this is the first, we study sufficient conditions for Hamiltonicity in terms of forbidden induced subgraphs and extend such results to locally finite infinite graphs. For this we use topological circles within the Freudenthal compactification of a locally finite graph as infinite cycles. In this paper we focus on conditions involving claws, nets and bulls as induced subgraphs. We extend Hamiltonicity results for finite claw-free and net-free graphs by Shepherd to locally finite graphs. Moreover, we generalise a classification of finite claw-free and net-free gra…
Let $G$ be a graph on $n$ vertices. An induced subgraph $H$ of $G$ is called heavy if there exist two nonadjacent vertices in $H$ with degree sum at least $n$ in $G$. We say that $G$ is $H$-heavy if every induced subgraph of $G$ isomorphic to $H$ is heavy. For a family $\mathcal{H}$ of graphs, $G$ is called $\mathcal{H}$-heavy if $G$ is $H$-heavy for every $H\in\mathcal{H}$. In this paper we characterize all connected graphs $R$ and $S$ other than $P_3$ (the path on three vertices) such that every 2-connected $\{R,S\}$-heavy graph is Hamiltonian. This extends several previous results on forbid…
A perfect matching reciprocity method for embedding multiple hypercubes in an augmented cube: Applications to Hamiltonian decomposition and fault-tolerant Hamiltonicity
Bedrossian characterized all pairs of forbidden subgraphs for a 2-connected graph to be Hamiltonian. Instead of forbidding some induced subgraphs, we relax the conditions for graphs to be Hamiltonian by restricting Ore- and Fan-type degree conditions on these induced subgraphs. Let $G$ be a graph on $n$ vertices and $H$ be an induced subgraph of $G$. $H$ is called \emph{o}-heavy if there are two nonadjacent vertices in $H$ with degree sum at least $n$, and is called $f$-heavy if for every two vertices $u,v\in V(H)$, $d_{H}(u,v)=2$ implies that $\max\{d(u),d(v)\}\geq n/2$. We say that $G$ is $H…
In 1984, Fan gave a sufficient condition involving maximum degree of every pair of vertices at distance two for a graph to be Hamiltonian. Motivated by Fan's result, we say that an induced subgraph $H$ of a graph $G$ is $f$-heavy if for every pair of vertices $u,v\in V(H)$, $d_{H}(u,v)=2$ implies that $\max\{d(u),d(v)\}\geq n/2$. For a given graph $R$, $G$ is called $R$-$f$-heavy if every induced subgraph of $G$ isomorphic to $R$ is $f$-heavy. For a family $\mathcal{R}$ of graphs, $G$ is $\mathcal{R}$-$f$-\emph{heavy} if $G$ is $R$-$f$-heavy for every $R\in \mathcal{R}$. In this note we show t…
Let claw be the graph $K_{1,3}$. A graph $G$ on $n\geq 3$ vertices is called \emph{o}-heavy if each induced claw of $G$ has a pair of end-vertices with degree sum at least $n$, and 1-heavy if at least one end-vertex of each induced claw of $G$ has degree at least $n/2$. In this note, we show that every 2-connected $o$-heavy or 3-connected 1-heavy graph is Hamiltonian if we restrict Fan-type degree condition or neighborhood intersection condition to certain pairs of vertices in some small induced subgraphs of the graph. Our results improve or extend previous results of Broersma et al., Chen et …
A graph $G$ is called \emph{claw-o-heavy} if every induced claw ($K_{1,3}$) of $G$ has two end-vertices with degree sum at least $|V(G)|$ in $G$. For a given graph $R$, $G$ is called \emph{$R$-f-heavy} if for every induced subgraph $H$ of $G$ isomorphic to $R$ and every pair of vertices $u,v\in V(H)$ with $d_H(u,v)=2$, there holds $\max\{d(u),d(v)\}\geq |V(G)|/2$. In this paper, we prove that every 2-connected claw-\emph{o}-heavy and $Z_3$-\emph{f}-heavy graph is hamiltonian (with two exceptional graphs), where $Z_3$ is the graph obtained from identifying one end-vertex of $P_4$ (a path with 4…
In this paper we modify slightly Razborov's flag algebra machinery to be suitable for the hypercube. We use this modified method to show that the maximum number of edges of a 4-cycle-free subgraph of the n-dimensional hypercube is at most 0.6068 times the number of its edges. We also improve the upper bound on the number of edges for 6-cycle-free subgraphs of the n-dimensional hypercube from the square root of 2 - 1 to 0.3755 times the number of its edges. Additionally, we show that if the n-dimensional hypercube is considered as a poset, then the maximum vertex density of three middle layers …
A graph is called \emph{claw-free} if it contains no induced subgraph isomorphic to $K_{1,3}$. Matthews and Sumner proved that a 2-connected claw-free graph $G$ is hamiltonian if every vertex of it has degree at least $(|V(G)|-2)/3$. At the workshop C\&C (Novy Smokovec, 1993), Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of $N$ (the graph obtained from a triangle by adding three disjoint pendant edges). Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of ind…
A classical extremal, or Turán-type problem asks to determine ${\rm ex}(G, H)$, the largest number of edges in a subgraph of a graph $G$ which does not contain a subgraph isomorphic to $H$. Alon and Shikhelman introduced the so-called generalized extremal number ${\rm ex}(G,T,H)$, defined to be the maximum number of subgraphs isomorphic to $T$ in a subgraph of $G$ that contains no subgraphs isomorphic to $H$. In this paper we investigate the case when $G = Q_n$, the hypercube of dimension $n$, and $T$ and $H$ are smaller hypercubes or cycles.
Let $Q_k$ denote the $k$-dimensional hypercube on $2^k$ vertices. A vertex in a subgraph of $Q_k$ is {\em full} if its degree is $k$. We apply the Kruskal-Katona Theorem to compute the maximum number of full vertices an induced subgraph on $n\leq 2^k$ vertices of $Q_k$ can have, as a function of $k$ and $n$. This is then used to determine $\min(\max(|V(H_1)|, |V(H_2)|))$ where (i) $H_1$ and $H_2$ are induced subgraphs of $Q_k$, and (ii) together they cover all the edges of $Q_k$, that is $E(H_1)\cup E(H_2) = E(Q_k)$.
Let $G$ be an induced subgraph of the hypercube $Q_k$ for some $k$. We show that if $|G|$ is a power of $2$ then, for sufficiciently large $n$, the vertex set of $Q_n$ can be partitioned into induced copies of $G$. This answers a question of Offner. In fact, we prove a stronger statement: if $X$ is a subset of $\{0,1\}^k$ for some $k$ and if $|X|$ is a power of $2$, then, for sufficiently large $n$, $\{0,1\}^n$ can be partitioned into isometric copies of $X$.
A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $n/2$ is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we extend Dirac's theorem in two directions and show that Dirac graphs are robustly Hamiltonian in a very strong sense. First, we consider a random subgraph of a Dirac graph obtained by taking each edge independently with probability $p$, and prove that there exists a constant $C$ such that if $p \ge C \lo…
Let $G$ be a graph. Adopting the terminology of Broersma et al. and Čada, respectively, we say that $G$ is 2-heavy if every induced claw ($K_{1,3}$) of $G$ contains two end-vertices each one has degree at least $|V(G)|/2$; and $G$ is o-heavy if every induced claw of $G$ contains two end-vertices with degree sum at least $|V(G)|$ in $G$. In this paper, we introduce a new concept, and say that $G$ is \emph{$S$-c-heavy} if for a given graph $S$ and every induced subgraph $G'$ of $G$ isomorphic to $S$ and every maximal clique $C$ of $G'$, every non-trivial component of $G'-C$ contains a vertex of …
Denote by $r_g(G,\mathcal{H})$ the global resilience of a graph $G$ with respect to Hamiltonicity. That is, $r_g(G,\mathcal{H})$ is the minimal $r$ for which there exists a subgraph $H\subseteq G$ with $r$ edges, such that $G\setminus H$ is not Hamiltonian. We show that if $p$ is above the Hamiltonicity threshold and $G\sim G(n,p)$ then, with high probability, $r_g(G,\mathcal{H})=δ(G)-1$. This is easily extended to the full interval: for every $p(n)\in [0,1]$, if $G\sim G(n,p)$ then, with high probability, $r_g(G,\mathcal{H})= \max \{ 0,δ(G)-1 \}$.
Given $p \in (0,1)$, we let $Q_p= Q_p^d$ be the random subgraph of the $d$-dimensional hypercube $Q^d$ where edges are present independently with probability $p$. It is well known that, as $d \rightarrow \infty$, if $p>\frac12$ then with high probability $Q_p$ is connected; and if $p<\frac12$ then with high probability $Q_p$ consists of one giant component together with many smaller components which form the `fragment'. Here we fix $p \in (0,\frac12)$, and investigate the fragment, and how it sits inside the hypercube. In particular we give asymptotic estimates for the mean numbers of componen…
Given a graph $G$, denote by $Δ$ and $χ^\prime$ the maximum degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$Δ$-critical} if $χ^\prime(G)=Δ+1$ and $χ^\prime(H)\leΔ$ for every proper subgraph $H$ of $G$. We proved that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{2n}{3}+12$ is Hamiltonian.
We prove that random hypergraphs are asymptotically almost surely resiliently Hamiltonian. Specifically, for any $γ>0$ and $k\ge3$, we show that asymptotically almost surely, every subgraph of the binomial random $k$-uniform hypergraph $G^{(k)}\big(n,n^{γ-1}\big)$ in which all $(k-1)$-sets are contained in at least $\big(\tfrac12+2γ\big)pn$ edges has a tight Hamilton cycle. This is a cyclic ordering of the $n$ vertices such that each consecutive $k$ vertices forms an edge.
Let $ex(Q_n, H)$ be the largest number of edges in a subgraph $G$ of a hypercube $Q_n$ such that there is no subgraph of $G$ isomorphic to $H$. We show that for any integer $k\geq 3$, $$ex(Q_n, C_{4k+2})= O(n^{\frac{5}{6} + \frac{1}{3(2k-2)}} 2^n).$$
The hypercube Q_n is the graph whose vertex set is {0,1}^n and where two vertices are adjacent if they differ in exactly one coordinate. For any subgraph H of the cube, let ex(Q_n, H) be the maximum number of edges in a subgraph of Q_n which does not contain a copy of H. We find a wide class of subgraphs H, including all previously known examples, for which ex(Q_n, H) = o(e(Q_n)). In particular, our method gives a unified approach to proving that ex(Q_n, C_{2t}) = o(e(Q_n)) for all t >= 4 other than 5.
The Fibonacci cube of dimension n, denoted as $Γ$ n , is the subgraph of n-cube Q n induced by vertices with no consecutive 1's. In this short note we prove that asymptotically all vertices of $Γ$ n are covered by a maximum set of disjoint subgraphs isomorphic to Q k , answering an open problem proposed in [2].
We say $G$ is \emph{$(Q_n,Q_m)$-saturated} if it is a maximal $Q_m$-free subgraph of the $n$-dimensional hypercube $Q_n$. A graph, $G$, is said to be $(Q_n,Q_m)$-semi-saturated if it is a subgraph of $Q_n$ and adding any edge forms a new copy of $Q_m$. The minimum number of edges a $(Q_n,Q_m)$-saturated graph (resp. $(Q_n,Q_m)$-semi-saturated graph) can have is denoted by $sat(Q_n,Q_m)$ (resp. $s\text{-}sat(Q_n,Q_m)$). We prove that $ \lim_{n\to\infty}\frac{sat(Q_n,Q_m)}{e(Q_n)}=0$, for fixed $m$, disproving a conjecture of Santolupo that, when $m=2$, this limit is $\frac{1}{4}$. Further, we s…
How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erdős about $27$ years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved. Let $f(n,C_l)$ be the largest number of edges in a subgraph of a hypercube $Q_n$ containing no cycle of length $l$. It is known that $f(n, C_l) = o(|E(Q_n)|)$, when $l= 4k$, $k\geq 2$ and that $f(n, C_6) \geq \frac{1}{3} |E(Q_n)|$. It is an open question to determine $f(n, C_l)$ for $l=4k+2$, $k\geq 2$. Here, we give a general upper bound for $f(n,C_l)$…
In this short note we study two questions about the existence of subgraphs of the hypercube $Q_n$ with certain properties. The first question, due to Erdős--Hamburger--Pippert--Weakley, asks whether there exists a bounded degree subgraph of $Q_n$ which has diameter $n$. We answer this question by giving an explicit construction of such a subgraph with maximum degree at most 120. The second problem concerns properties of $k$-additive spanners of the hypercube, that is, subgraphs of $Q_n$ in which the distance between any two vertices is at most $k$ larger than in $Q_n$. Denoting by $Δ_{k,\infty…
Let $G$ be a graph and $T$ a certain connected subgraph of $G$. The $T$-structure connectivity $κ(G; T)$ (or resp., $T$-substructure connectivity $κ^{s}(G; T)$) of $G$ is the minimum number of a set of subgraphs $\mathcal{F}=\{T_{1}, T_{2}, \ldots, T_{m}\}$ (or resp., $\mathcal{F}=\{T^{'}_{1}, T^{'}_{2}, \ldots, T^{'}_{m}\}$) such that $T_{i}$ is isomorphic to $T$ (or resp., $T^{'}_{i}$ is a connected subgraph of $T$) for every $1\leq i \leq m$, and $\mathcal{F}$'s removal will disconnect $G$. The twisted hypercube $H_{n}$ is a new variant of hypercubes with asymptotically optimal diameter int…
Let $Q_{d,p}$ be the random subgraph of the $d$-dimensional hypercube $\{0,1\}^d$, where each edge is retained independently with probability $p$. We study the asymptotic number of independent sets in $Q_{d,p}$ as $d \to \infty$ for a wide range of parameters $p$, including values of $p$ tending to zero as fast as $\frac{C\log d}{d^{1/3}}$, constant values of $p$, and values of $p$ tending to one. The results extend to the hardcore model on $Q_{d,p}$, and are obtained by studying the closely related antiferromagnetic Ising model on the hypercube, which can be viewed as a positive-temperature h…
In this paper, we show that every $(2^{n-1}+1)$-vertex induced subgraph of the $n$-dimensional cube graph has maximum degree at least $\sqrt{n}$. This result is best possible, and improves a logarithmic lower bound shown by Chung, Füredi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.
Let G be a random subgraph of the n-cube where each edge appears randomly and independently with probability p. We prove that the largest eigenvalue of the adjacency matrix of G is almost surely λ_1(G)= (1+o(1)) max(Δ^{1/2}(G),np), where Δ(G) is the maximum degree of G and o(1) term tends to zero as max (Δ^{1/2}(G), np) tends to infinity.
The Fibonacci cube $Γ_n$ is the subgraph of the hypercube induced by the binary strings that contain no two consecutive 1's. The Lucas cube $Λ_n$ is obtained from $Γ_n$ by removing vertices that start and end with 1. We characterize maximal induced hypercubes in $Γ_n$ and $Λ_n$ and deduce for any $p\leq n$ the number of maximal $p$-dimensional hypercubes in these graphs.
A graph is Hamiltonian if it contains a cycle which visits every vertex of the graph exactly once. In this paper, we consider the problem of Hamiltonicity of a graph $G_n$, which will be called the prime difference graph of order $n$, with vertex set $\{1,2,\cdots, n\}$ and edge set $\{uv: |u-v|$ is a prime number$\}$. A recent result, conjectured by Sun and later proved by Chen, asserts that $G_n$ is Hamiltonian for $n\geq 5$. This paper extends their result in three directions. First, we prove that for any two integers $a$ and $b$ with $1\leq a<b\leq n$, there is a Hamilton path in $G_n$ fro…
The {\em Fibonacci cube} of dimension $n$, denoted as $Γ\_n$, is the subgraph of $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. We study the maximum number of disjoint subgraphs in $Γ\_n$ isomorphic to $Q\_k$, and denote this number by $q\_k(n)$. We prove several recursive results for $q\_k(n)$, in particular we prove that $q\_{k}(n) = q\_{k-1}(n-2) + q\_{k}(n-3)$. We also prove a closed formula in which $q\_k(n)$ is given in terms of Fibonacci numbers, and finally we give the generating function for the sequence $\{q\_{k}(n)\}\_{n=0}^{ \infty}$.
Let $\mathcal{G}(k)$ denote the set of connected $k$-regular graphs $G$, $k\geq2$, where the number of vertices at distance 2 from any vertex in $G$ does not exceed $k$. Asratian (2006) showed (using other terminology) that a graph $G\in\mathcal{G}(k)$ is Hamiltonian if for each vertex $u$ of $G$ the subgraph induced by the set of vertices at distance at most 2 from $u$ is 2-connected. We prove here that in fact all graphs in the sets $\mathcal{G}(3)$, $\mathcal{G}(4)$ and $\mathcal{G}(5)$ are Hamiltonian. We also prove that the problem of determining whether there exists a Hamilton cycle in a…
Let $H$ be an induced subgraph of the hypercube $Q_k$, for some $k$. We show that for some $c = c(H)$, the vertices of $Q_n$ can be partitioned into induced copies of $H$ and a remainder of at most $O(n^c)$ vertices. We also show that the error term cannot be replaced by anything smaller than $\log n$.
We consider a sparse random subraph of the $n$-cube where each edge appears independently with small probability $p(n) =O(n^{-1+o(1)})$. In the most interesting regime when $p(n)$ is not exponentially small we prove that the largest eigenvalue of the graph is asymtotically equal to the square root of the maximum degree.
Recently, Huang gave a very elegant proof of the Sensitivity Conjecture by proving that hypercube graphs have the following property: every induced subgraph on a set of more than half its vertices has maximum degree at least $\sqrt{d}$, where $d$ is the valency of the hypercube. This was generalised by Alon and Zheng who proved that every Cayley graph on an elementary abelian $2$-group has the same property. Very recently, Potechin and Tsang proved an analogous results for Cayley graphs on abelian groups. They also conjectured that all Cayley graphs have the analogous property. We disprove thi…
If $H$ is (or is isomorphic to) a subgraph of $G$, $H$ is said to {\it divide} $G$ if there is an edge-decomposition of $G$ by copies of $E(H)$, the edge set of $H$. A more restrictive version of this is when there is a subgroup ${\cal H}$ of {\rm Aut} $(G)$, the automorphism group of $G$, such that the copies of $E(H)$ are the translates of $E(H)$ by the elements of ${\cal H}$. In a paper by the second author, this situation was described by saying that $H$, or more precisely $E(H)$, is a {\it fundamental} set for $G$. Many authors have studied the notion of divisibility for various graphs, a…
The hypercube of dimension n is the graph whose vertices are the 2^n binary words of length n, and there is an edge between two of them if they have Hamming distance 1. We consider an edit distance based on swaps and mismatches, to which we refer as tilde-distance, and define the tilde-hypercube with edges linking words at tilde-distance 1. Then, we introduce and study some isometric subgraphs of the tilde-hypercube obtained by using special words called tilde-isometric words. The subgraphs keep only the vertices that avoid a given tilde-isometric word as a factor. In the case of word 11, the …
In 1981, Duffus, Gould, and Jacobson showed that every connected graph either has a Hamiltonian path, or contains a claw ($K_{1,3}$) or a net (a fixed six-vertex graph) as an induced subgraph. This implies that subject to being connected, these two are the only minimal (under taking induced subgraphs) graphs with no Hamiltonian path. Brousek (1998) characterized the minimal graphs that are $2$-connected, non-Hamiltonian and do not contain the claw as an induced subgraph. We characterize the minimal graphs that are $2$-connected and non-Hamiltonian for two classes of graphs: (1) split graphs, (…
Analogous to the case of the binomial random graph $G(d+1,p)$, it is known that the behaviour of a random subgraph of a $d$-dimensional hypercube, where we include each edge independently with probability $p$, which we denote by $Q^d_p$, undergoes a phase transition around the critical value of $p=\frac{1}{d}$. More precisely, standard arguments show that significantly below this value of $p$, with probability tending to one as $d \to \infty$ (whp for short) all components of this graph have order $O(d)$, whereas Ajtai, Komlós and Szemerédi showed that significantly above this value, in the \e…
The connectivity of a network directly signifies its reliability and fault-tolerance. Structure and substructure connectivity are two novel generalizations of the connectivity. Let $H$ be a subgraph of a connected graph $G$. The structure connectivity (resp. substructure connectivity) of $G$, denoted by $κ(G;H)$ (resp. $κ^s(G;H)$), is defined to be the minimum cardinality of a set $F$ of connected subgraphs in $G$, if exists, whose removal disconnects $G$ and each element of $F$ is isomorphic to $H$ (resp. a subgraph of $H$). In this paper, we shall establish both $κ(BH_n;H)$ and $κ^s(BH_n;H)$…
Many authors have investigated edge decompositions of graphs by the edge sets of isomorphic copies of special subgraphs. For $q$- dimensional hypercubes $Q_q$ various researchers have done this for cer- tain trees, paths, and cycles. In this paper we shall say that "$H$ divides $G$" if $E(G)$ is the disjoint union of $\{fE(H_i) | H_i \simeq H\}$. Our main result is that for $q$ odd and $q < 2^{32}$, the path of length $m$, $P_m$, divides $Q_q$ if and only if $m \leq q$ and $m | (q \times 2^{q-1})$.
It is well-known that the behaviour of a random subgraph of a $d$-dimensional hypercube, where we include each edge independently with probability $p$, undergoes a phase transition when $p$ is around $\frac{1}{d}$. More precisely, standard arguments show that just below this value of $p$ all components of this graph have order $O(d)$ with probability tending to one as $d \to \infty$ (whp for short), whereas Ajtai, Komlós and Szemerédi [Largest random component of a $k$-cube, Combinatorica 2 (1982), no. 1, 1--7; MR0671140] showed that just above this value, in the supercritical regime, whp ther…
Let $\{G_M\}_{M\geq 0}$ be the random graph process, where $G_0$ is the empty graph on $n$ vertices and subsequent graphs in the sequence are obtained by adding a new edge uniformly at random. For each $\varepsilon>0$, we show that, almost surely, any graph $G_M$ with minimum degree at least 2 is not only Hamiltonian (as shown by Bollobás), but remains Hamiltonian despite the removal of any set of edges, as long as at most $(1/2-\varepsilon)$ of the edges incident to each vertex are removed. We say that such a graph is $(1/2-\varepsilon)$-resiliently Hamiltonian. Furthermore, for each $ε>0$, w…
A path in the hypercube $Q_n$ is said to be a geodesic if no two of its edges are in the same direction. Let $G$ be a subgraph of $Q_n$ with average degree $d$. How long a geodesic must $G$ contain? We show that $G$ must contain a geodesic of length $d$. This result, which is best possible, strengthens a theorem of Feder and Subi. It is also related to the `antipodal colourings' conjecture of Norine.
We introduce a notion of the \emph{crux} of a graph $G$, measuring the order of a smallest dense subgraph in $G$. This simple-looking notion leads to some generalisations of known results about cycles, offering an interesting paradigm of `replacing average degree by crux'. In particular, we prove that \emph{every} graph contains a cycle of length linear in its crux. Long proved that every subgraph of a hypercube $Q^m$ (resp. discrete torus $C_3^m$) with average degree $d$ contains a path of length $2^{d/2}$ (resp. $2^{d/4}$), and conjectured that there should be a path of length $2^{d}-1$ (res…
A graph $H^{\prime}$ is $(H, G)$-saturated if it is $G$-free and the addition of any edge of $H$ not in $H^{\prime}$ creates a copy of $G$. The saturation number $sat(H, G)$ is the minimum number of edges in a $(H, G)$-saturated graph. We investigate bounds on the saturation number of trees $T$ in the $n$-dimensional hypercube $Q_n$. We first present a general lower bound on the saturation number based on the minimum degree of non-leaves. From there, we suggest two general methods for constructing $T$-saturated subgraphs of $Q_n$, and prove nontrivial upper bounds for specific types of trees, …
Given a graph $Γ= (V, E)$ on $n$ vertices and $m$ edges, we define the Erdős-Rényi graph process with host $Γ$ as follows. A permutation $e_1,\dots,e_m$ of $E$ is chosen uniformly at random, and for $t\leq m$ we let $Γ_t = (V, \{e_1,\dots,e_t\})$. Suppose the minimum degree of $Γ$ is $δ(Γ) \geq (1/2 + \varepsilon)n$ for some constant $\varepsilon > 0$. Then with high probability, $Γ_t$ becomes Hamiltonian at the same moment that its minimum degree becomes at least two. Given $0\leq p\leq 1$ we let $Γ_p$ be the Erdős-Rényi subgraph of $Γ$, obtained by retaining each edge independently with prob…
