Hamiltonicity of the Double Vertex Graph and the Complete Double Vertex Graph of some Join Graphs
Luis Enrique Adame, Luis Manuel Rivera, Ana Laura Trujillo-Negrete
备注：
V2 is a revised version. Not intended for publication. Theorem 1.1 was presented in Symmetry 13(6) (1076), 2021, doi:10.3390/sym13061076 (arXiv:2101.01855) and Theorem 1.2 was presented in arXiv:2108.01119 (to appear in Matemática Contemporânea)
Let $G$ be a simple graph of order $n$. The double vertex graph $F_2(G)$ of $G$ is the graph whose vertices are the $2$-subsets of $V(G)$, where two vertices are adjacent in $F_2(G)$ if their symmetric difference is a pair of adjacent vertices in $G$. A generalization of this graph is the complete double vertex graph $M_2(G)$ of $G$, defined as the graph whose vertices are the $2$-multisubsets of $V(G)$, and two of such vertices are adjacent in $M_2(G)$ if their symmetric difference (as multisets) is a pair of adjacent vertices in $G$. In this paper we exhibit an infinite family of graphs (containing Hamiltonian and non-Hamiltonian graphs) for which $F_2(G)$ and $M_2(G)$ are Hamiltonian. This family of graphs is the set of join graphs $G=G_1 + G_2$, where $G_1$ and $G_2$ are of order $m\geq 1$ and $n\geq 2$, respectively, and $G_2$ has a Hamiltonian path. For this family of graphs, we show that if $m\leq 2n$ then $F_2(G)$ is Hamiltonian, and if $m\leq 2(n-1)$ then $M_2(G)$ is Hamiltonian.
Given a graph $H$, let $G^j_k(H)$ be the graph whose vertices are the proper $k$-colorings of $H$, with edges joining two colorings if $H$ contains a connected subgraph on at most $j$ vertices that includes all vertices where the colorings differ. Properties of $G^1_k(H)$ have been investigated before, including connectedness and Hamiltonicity. We introduce and study the parameters $g_k(H)$ and $h_k(H)$, which denote the minimum $j$ such that $G^j_k(H)$ is connected or Hamiltonian, respectively.
The complete double vertex graph $M_2(G)$ of $G$ is defined as the graph whose vertices are the $2$-multisubsets of $V(G)$, and two of such vertices are adjacent in $M_2(G)$ if their symmetric difference (as multisets) is a pair of adjacent vertices in $G$. In this paper we exhibit an infinite family of graphs $G$ (containing Hamiltonian and non-Hamiltonian graphs) for which $M_2(G)$ are Hamiltonian.
It is shown that every connected vertex-transitive graph of order $4p$, where $p$ is a prime, is hamiltonian with the exception of the Coxeter graph which is known to possess a Hamilton path.
For any undirected and simple graph G = (V;E), where V denotes the vertex set and E the edge set of G. G is called hamiltonian if it contains a cycle that visits each vertex of G exactly once. Ore (1960) proved that G is hamiltonian if degG(u) + degG(v) >= n holds for every nonadjacent pair of vertices u and v in V, where n is the total number of distinct vertices of G. Kao et al. (2012) proved that any graph G satisfying Ore's theorem remains hamiltonian after the removal of any vertex x in V unless G belongs to one of the two exceptional families of graphs. In this paper, we proved that in f…
A graph $G$ is said to be $k$-$γ$-vertex critical if the domination numbers $γ(G)$ of $G$ is $k$ and $γ(G - v) < k$ for any vertex $v$ of $G$. Similarly, A graph $G$ is said to be $k$-$γ_{c}$-vertex critical if the connected domination numbers $γ_{c}(G)$ of $G$ is $k$ and $γ_{c}(G - v) < k$ for any vertex $v$ of $G$. The problem of interest is to determine whether or not $2$-connected $k$-$γ$-vertex critical graphs are Hamiltonian. In this paper, for all $k \geq 3$, we provide a $2$-connected $k$-$γ$-vertex critical graph which is non-Hamiltonian. We prove that every $2$-connected $3$-$γ$-vert…
We show that for any fixed $α>0$, cherry-quasirandom 3-graphs of positive density and sufficiently large order $n$ with minimum vertex degree $α\binom n2$ have a tight Hamilton cycle. This solves a conjecture of Aigner-Horev and Levy.
Let $\vec C_m$ and $\vec C_n$ be directed cycles of length $m$ and $n$, with $m,n \ge 3$, and let $P(\vec C_m \mathbin{\Box} \vec C_n)$ be the digraph that is obtained from the Cartesian product $\vec C_m \mathbin{\Box} \vec C_n$ by choosing a vertex $v$, and reversing the orientation of all four directed edges that are incident with $v$. (This operation is called "pushing" at the vertex $v$.) By applying a special case of unpublished work of S.X.Wu, we find elementary number-theoretic necessary and sufficient conditions for the existence of a hamiltonian cycle in $P(\vec C_m \mathbin{\Box} \v…
A graph is Hamiltonian if it contains a cycle which goes through all vertices exactly once. Determining if a graph is Hamiltonian is known as a NP-complete problem and no satisfactory characterization for these graphs has been found. In 1976 Bondy and Chvatal introduced a way to get round the Hamiltonicity problem complexity by using a closure of the graph. This closure is a supergraph of G which preserves Hamiltonicity, that is, which is Hamiltonian if and only if G is. Since this seminal work, several closure concepts preserving Hamiltonicity were introduced. In particular Ryjacek defined in…
We study Hamiltonicity in graphs obtained as the union of a deterministic $n$-vertex graph $H$ with linear degrees and a $d$-dimensional random geometric graph $G^d(n,r)$, for any $d\geq1$. We obtain an asymptotically optimal bound on the minimum $r$ for which a.a.s. $H\cup G^d(n,r)$ is Hamiltonian. Our proof provides a linear time algorithm to find a Hamilton cycle in such graphs.
For integers $k\ge 3$ and $1\le \ell\le k-1$, we prove that for any $α>0$, there exist $ε>0$ and $C>0$ such that for sufficiently large $n\in (k-\ell)\mathbb{N}$, the union of a $k$-uniform hypergraph with minimum vertex degree $αn^{k-1}$ and a binomial random $k$-uniform hypergraph $\mathbb{G}^{(k)}(n,p)$ with $p\ge n^{-(k-\ell)-ε}$ for $\ell\ge 2$ and $p\ge C n^{-(k-1)}$ for $\ell=1$ on the same vertex set contains a Hamiltonian $\ell$-cycle with high probability. Our result is best possible up to the values of $ε$ and $C$ and answers a question of Krivelevich, Kwan and Sudakov.
In this paper, we study the Hamiltonicity of graphs with large minimum degree. Firstly, we present some conditions for a simple graph to be Hamilton-connected and traceable from every vertex in terms of the spectral radius of the graph or its complement respectively. Secondly, we give the conditions for a nearly balanced bipartite graph to be traceable in terms of spectral radius, signless Laplacian spectral radius of the graph or its quasi-complement respectively.
Let $G$ be a simple graph of order $n$ and let $k$ be an integer such that $1\leq k\leq n-1$. The $k$-token graph $G^{\{k\}}$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $G^{\{k\}}$ whenever their symmetric difference is a pair of adjacent vertices in $G$. In this paper we study the Hamiltonicity of the $k$-token graphs of some join graphs. As a consequence, we provide an infinite family of graphs (containing Hamiltonian and non-Hamiltonian graphs) for which their $k$-token graphs are Hamiltonian. Our result provides, to our knowledge, t…
R. Wang (Discrete Mathematics and Theoretical Computer Science, vol. 19(3), 2017) proposed the following problem. \textbf{Problem.} Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 8$. Suppose that $d(x)\geq 2a-k$, $ d(y)\geq a+k$ or $d(y)\geq 2a-k$, $ d(x)\geq a+k$ for every pair of vertices $x,y$ with a common out-neighbour, where $2 \leq k\leq a/2$. Is $D$ Hamiltonian? In this paper, we prove that if a digraph $D$ satisfies the conditions of this problem, then (i) $D$ contains a cycle factor, (ii) for every vertex $x\in V(D)$ there exists a vertex $y\in V(D…
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…
A generalized de Bruijn digraph generalizes a de Bruijn digraph to the case where the number of vertices need not be a pure power of an integer. Hamiltonian cycles in these digraphs thus generalize regular de~Bruijn cycles, and we will thus refer to them simply as de Bruijn cycles. We define the cross-join to be the graph with all de Bruijn cycles as vertices, there is an edge between two of these vertices if one can be obtained from the other via a cross-join operation. We show that the cross-join graph is connected. This in particular means that any regular de Bruijn cycle can be cross-joine…
The prism over a graph $G$ is the cartesian product $G \Box K_2$. It is known that the property of having a Hamiltonian prism (prism-Hamiltonicity) is stronger than that of having a $2$-walk (spanning closed walk using every vertex at most twice) and weaker than that of having a Hamilton path. For a graph $G$, it is known that $α(G) \leq 2 κ(G)$, where $α(G)$ is the independence number and $κ(G)$ is the connectivity, imples existence of a $2$-walk in $G$, and the bound is sharp. West asked for a bound on $α(G)$ in terms of $κ(G)$ guaranteeing prism-Hamiltonicity. In this paper we answer this q…
In this paper, we discuss the Hamiltonicity of graphs in terms of Wiener index, hyper-Wiener index and Harary index of their quasi-complement or complement. Firstly, we give some sufficient conditions for an balanced bipartite graph with given the minimum degree to be traceable and Hamiltonian, respectively. Secondly, we present some sufficient conditions for a nearly balanced bipartite graph with given the minimum degree to be traceable. Thirdly, we establish some conditions for a graph with given the minimum degree to be traceable and Hamiltonian, respectively. Finally, we provide some condi…
An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two arcs $uv, xy$ are adjacent if and only if $(v,u,x,y)$ is a 3-arc of $G$. In this paper we prove that any connected 3-arc graph is Hamiltonian, and all iterative 3-arc graphs of any connected graph of minimum degree at least three are Hamiltonian. As a consequence we obtain that if a vertex-transitive graph is isomorphic to the 3-arc graph of a connected arc-…
The \emph{prism} over a graph $G$ is the product $G \Box K_2$, i.e., the graph obtained by taking two copies of $G$ and adding a perfect matching joining the two copies of each vertex by an edge. The graph $G$ is called \emph{prism-hamiltonian} if it has a hamiltonian prism. Jung showed that every $1$-tough $P_4$-free graph with at least three vertices is hamiltonian. In this paper, we extend this to observe that for $k \geq 1$ a $P_4$-free graph has a spanning \emph{$k$-walk} (closed walk using each vertex at most $k$ times) if and only if it is $\frac{1}{k}$-tough. As our main result, we sho…
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…
The $k$-coprime graph of order $n$ is the graph with vertex set $\{k, k+1, \ldots, k+n-1\}$ in which two vertices are adjacent if and only if they are coprime. We characterize Hamiltonian $k$-coprime graphs. As a particular case, two conjectures by Tout, Dabboucy, Howalla (1982) and by Schroeder (2019) on prime labeling of $2$-regular graphs follow. A prime labeling of a graph with $n$ vertices is a labeling of its vertices with distinct integers from $\{1, 2,\ldots , n\}$ in such a way that the labels of any two adjacent vertices are relatively prime.
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…
Let $\{D_M\}_{M\geq 0}$ be the $n$-vertex random directed graph process, where $D_0$ is the empty directed graph on $n$ vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each $\varepsilon>0$, we show that, almost surely, any directed graph $D_M$ with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most $(1/2-\varepsilon)$ of both the in- and out-edges incident to each vertex are removed. We say such a directed grap…
Dirac's theorem (1952) is a classical result of graph theory, stating that an $n$-vertex graph ($n \geq 3$) is Hamiltonian if every vertex has degree at least $n/2$. Both the value $n/2$ and the requirement for every vertex to have high degree are necessary for the theorem to hold. In this work we give efficient algorithms for determining Hamiltonicity when either of the two conditions are relaxed. More precisely, we show that the Hamiltonian cycle problem can be solved in time $c^k \cdot n^{O(1)}$, for some fixed constant $c$, if at least $n-k$ vertices have degree at least $n/2$, or if all v…
In his seminal paper from 1952 Dirac showed that the complete graph on $n\geq 3$ vertices remains Hamiltonian even if we allow an adversary to remove $\lfloor n/2\rfloor$ edges touching each vertex. In 1960 Ghouila-Houri obtained an analogue statement for digraphs by showing that every directed graph on $n\geq 3$ vertices with minimum in- and out-degree at least $n/2$ contains a directed Hamilton cycle. Both statements quantify the robustness of complete graphs (digraphs) with respect to the property of containing a Hamilton cycle. A natural way to generalize such results to arbitrary graphs (…
We study Hamiltonicity and pancyclicity in the graph obtained as the union of a deterministic $n$-vertex graph $H$ with $δ(H)\geqαn$ and a random $d$-regular graph $G$, for $d\in\{1,2\}$. When $G$ is a random $2$-regular graph, we prove that a.a.s. $H\cup G$ is pancyclic for all $α\in(0,1]$, and also extend our result to a range of sublinear degrees. When $G$ is a random $1$-regular graph, we prove that a.a.s. $H\cup G$ is pancyclic for all $α\in(\sqrt{2}-1,1]$, and this result is best possible. Furthermore, we show that this bound on $δ(H)$ is only needed when $H$ is `far' from containing a p…
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 pairing of a graph $G$ is a perfect matching of the complete graph having the same vertex set as $G$. If every pairing of $G$ can be extended to a Hamiltonian cycle of the underlying complete graph using only edges from $G$, then $G$ has the PH-property. A somewhat weaker property is the PMH-property, whereby every perfect matching of $G$ can be extended to a Hamiltonian cycle of $G$. In an attempt to characterise all 4-regular graphs having the PH-property, we answer a question made in 2015 by Alahmadi et al. by showing that the Cartesian product $C_p\square C_q$ of two cycles on $p$ and $q…
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 …
An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with given two different vertices s, t and an edge e to have (1)a Hamiltonian s-path, (2) a Hamiltonian st-path, (3) a Hamiltonian s- and st-paths containing edge e when G has connectivity one, and (4) a Hamiltonian cycle containing e when G is 2-connected. These results imply that a connected {claw, net}-free graph has a Hamiltonian path and a 2-connected {claw, net}-free graph has a Hamiltonian cycle…
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…
A step forward is made in a long standing Lovász's problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes, other than the Petersen graph, contains a Hamilton cycle. Essential tools used in the proof range from classical results on existence of Hamilton cycles, such as Chvátal's theorem and Jackson's theorem, to certain results on polynomial representations of quadratic residues at primitive roots in finite fields.
Let $\{G_i\}$ be the random graph process: starting with an empty graph $G_0$ with $n$ vertices, in every step $i \geq 1$ the graph $G_i$ is formed by taking an edge chosen uniformly at random among the non-existing ones and adding it to the graph $G_{i - 1}$. The classical `hitting-time' result of Ajtai, Komlós, and Szemerédi, and independently Bollobás, states that asymptotically almost surely the graph becomes Hamiltonian as soon as the minimum degree reaches $2$, that is if $δ(G_i) \ge 2$ then $G_i$ is Hamiltonian. We establish a resilience version of this result. In particular, we show th…
An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. A celebrated meta-conjecture of Bondy states that every non-trivial condition implying Hamiltonicity also implies pancyclicity (up to possibly a few exceptional graphs). We show that every graph $G$ with $κ(G) > (1+o(1)) α(G)$ is pancyclic. This extends the famous Chvátal-Erdős condition for Hamiltonicity and proves asymptotically a $30$-year old conjecture of Jackson and Ordaz.
We show that under certain conditions the square of the graph obtained by identifying a vertex in two graphs with hamiltonian square is also hamiltonian. Using this result, we prove necessary and sufficient conditions for hamiltonicity of the square of a connected graph such that every vertex of degree at least three in a block graph corresponds to a cut vertex and any two these vertices are at distance at least four.
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…
Following a problem posed by Lovász in 1969, it is believed that every connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from groups having a $(2,s,3)$-presentation, that is, for groups $G=\la a,b| a^2=1, b^s=1, (ab)^3=1, etc. \ra$ generated by an involution $a$ and an element $b$ of order $s\geq3$ such that their product $ab$ has order 3. More precisely, it is shown that the Cayley graph $X=Cay(G,\{a,b,b^{-1}\})$ has a Hamilton cycle when $|G|$ (and thus $s$) is congruent to 2 modulo 4, and has a long cycle missing only two ve…
We consider the random graph $G_{n, {\bf d}}$ chosen uniformly at random from the set of all graphs with a given sparse degree sequence ${\bf d}$. We assume ${\bf d}$ has minimum degree at least 4, at most a power law tail, and place one more condition on its tail. For $k\ge 2$ define $β_k(G) = \max e(A, B) + k(|A|-|B|) - d(A)$, with the maximum taken over disjoint vertex sets $A, B$. It is shown that the problem of determining if $G_{n, {\bf d}}$ contains a Hamilton cycle reduces to calculating $β_2(G_{n, {\bf d}})$. If $k\ge 2$ and $δ\ge k+2$, the problem of determining if $G_{n, {\bf d}}$ c…
In this paper, we investigate some relations between the invariants (including vertex and edge connectivity and forwarding indices) of a graph and its Laplacian eigenvalues. In addition, we present a sufficient condition for the existence of Hamiltonicity in a graph involving its Laplacian eigenvalues.
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 …
Let $X_1,..., X_n$ be independent, uniformly random points from $[0,1]^2$. We prove that if we add edges between these points one by one by order of increasing edge length then, with probability tending to 1 as the number of points $n$ tends to $\infty$, the resulting graph gets its first Hamilton cycle at exactly the same time it loses its last vertex of degree less than two. This answers an open question of Penrose and provides an analogue for the random geometric graph of a celebrated result of Ajtai, Komlós and Szemerédi and independently of Bollobás on the usual random graph. We are also …
Let $G$ be an $n$-vertex graph with $n\ge 3$. A classic result of Dirac from 1952 asserts that $G$ is hamiltonian if $δ(G)\ge n/2$. Dirac's theorem is one of the most influential results in the study of hamiltonicity and by now there are many related known results\,(see, e.g., J. A. Bondy, Basic Graph Theory: Paths and Circuits, Chapter 1 in: {\it Handbook of Combinatorics Vol.1}). A {\it Halin graph} is a planar graph consisting of two edge-disjoint subgraphs: a spanning tree of at least 4 vertices and with no vertex of degree 2, and a cycle induced on the set of the leaves of the spanning tr…
