The relation between Hamiltonicity and toughness of a graph is a long standing research problem. The paper studies the Hamiltonicity of the Cartesian product graph $G_1\square G_2$ of graphs $G_1$ and $G_2$ satisfying that $G_1$ is traceable and $G_2$ is connected with a path factor. Let Pn be the path of order $n$ and $H$ be a connected bipartite graph. With certain requirements of $n$, we show that the following three statements are equivalent: (i) $P_n\square H$ is Hamiltonian; (ii) $P_n\square H$ is $1$-tough; and (iii) $H$ has a path factor.
Chen et al. proved that every 18-tough chordal graph has a Hamilton cycle [Networks 31 (1998), 29-38]. Improving upon their bound, we show that every 10-tough chordal graph is Hamiltonian (in fact, Hamilton-connected). We use Aharoni and Haxell's hypergraph extension of Hall's Theorem as our main tool.
A graph is called $2K_2$-free if it does not contain two independent edges as an induced subgraph. Broersma, Patel, and Pyatkin showed that every 25-tough $2K_2$-free graph with at least three vertices is hamiltonian. In this paper, we improve the required toughness in this result from 25 to 3.
The toughness of a noncomplete graph $G$ is the maximum real number $t$ such that the ratio of $|S|$ to the number of components of $G-S$ is at least $t$ for every cutset $S$ of $G$, and the toughness of a complete graph is defined to be $\infty$. Determining the toughness for a given graph is NP-hard. Chvátal's toughness conjecture, stating that there exists a constant $t_0$ such that every graph with toughness at least $t_0$ is hamiltonian, is still open for general graphs. A graph is called $(P_3\cup 2P_1)$-free if it does not contain any induced subgraph isomorphic to $P_3\cup 2P_1$, the d…
Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. Determining toughness is an NP-hard problem for arbitrary graphs. The Toughness Conjecture of Chvátal, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is hamiltonian, is still open in general. A graph is called $(P_2\cup P_3)$-free if it does not contain any induced subgraph isomorphic to $P_2\cup P_3$, the union of two vertex-disjoint paths of order 2 and 3, resp…
Let $k \geq 2$ be an integer. We say that a graph $G$ is $(K_2 \cup kK_1)$-free if it does not contain $K_2 \cup kK_1$ as an induced subgraph. Recently, Shi and Shan conjectured that every $1$-tough and $2k$-connected $(K_2 \cup kK_1)$-free graph is hamiltonian. In this paper, we solve this conjecture by proving the statement; every $1$-tough and $k$-connected $(K_2 \cup kK_1)$-free graph with minimum degree at least $\frac{3(k-1)}{2}$ is hamiltonian or the Petersen graph.
Studying the shortness of longest cycles in maximal planar graphs, we improve the upper bound on the shortness exponent of the class of $\frac{5}{4}$-tough maximal planar graphs presented by Harant and Owens [Discrete Math. 147 (1995), 301--305]. In addition, we present two generalizations of a similar result of Tkáč who considered $1$-tough maximal planar graphs [Discrete Math. 154 (1996), 321--328]; we remark that one of these generalizations gives a tight upper bound. We fix a problematic argument used in the first paper.
A graph $G$ is called a $2K_2$-free graph if it does not contain $2K_2$ as an induced subgraph. In 2014, Broersma, Patel and Pyatkin showed that every 25-tough $2K_2$-free graph on at least three vertices is Hamiltonian. Recently, Shan improved this result by showing that 3-tough is sufficient instead of 25-tough. In this paper, we show that every 2-tough $2K_2$-free graph on at least three vertices is Hamiltonian, which was conjectured by Gao and Pasechnik.
In 1974, Goodman and Hedetniemi proved that every 2-connected $(K_{1,3},K_{1,3}+e)$-free graph is hamiltonian. This result gave rise many other hamiltonicity conditions for various pairs and triples of forbidden connected subgraphs under additional connectivity conditions. In 1997, it was proved that a single forbidden connected subgraph $R$ in 2-connected graphs can create only a trivial class of hamiltonian graphs (complete graphs) with $R=P_3$. In this paper we prove that a single forbidden subgraph $R$ can create a non trivial class of hamiltonian graphs if $R$ is disconnected: $(\ast1)$ e…
Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. The Toughness Conjecture of Chvátal, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is hamiltonian, is still open in general. For any given integer $k\ge 1$, a graph $G$ is $(P_2\cup kP_1)$ free if $G$ does not contain the disjoint union of $P_2$ and $k$ isolated vertices as an induced subgraph. In this note, we show that every 4-tough and $2k$-connected $(P_2\cu…
Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. For a positive real number $t$, a non-complete graph $G$ is said to be $t$-tough if for every vertex cut $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. A complete graph is said to be $t$-tough for any $t>0$. Chvátal's toughness conjecture, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is Hamiltonian, is still open in general. Chvátal and Erdös \cite{CE} proved that, for any integer $k\ge 1$, every $\max\…
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…
The study of the existence of hamiltonian cycles in a graph is a classic problem in graph theory. By incorporating toughness and spectral conditions, we can consider Chvátal's conjecture from another perspective: what is the spectral condition to guarantee the existence of a hamiltonian cycle among $t$-tough graphs? We first give the answer to $1$-tough graphs, i.e. if $ρ(G)\geqρ(M_{n})$, then $G$ contains a hamiltonian cycle, unless $G\cong M_{n}$, where $M_{n}=K_{1}\nabla K_{n-4}^{+3}$ and $K_{n-4}^{+3}$ is the graph obtained from $3K_{1}\cup K_{n-4}$ by adding three independent edges betwee…
In 1952, Dirac proved that every 2-connected graph with minimum degree $δ$ either is hamiltonian or contains a cycle of length at least $2δ$. In 1986, Bauer and Schmeichel enlarged the bound $2δ$ to $2δ+2$ under additional 1-tough condition - an alternative and more natural necessary condition for a graph to be hamiltonian. In fact, the bound $2δ+2$ is sharp for a graph on $n$ vertices when $n\equiv 1(mod\ 3)$. In this paper we present the final version of this result which is sharp for each $n$: every 1-tough graph either is hamiltonian or contains a cycle of length at least $2δ+2$ when $n\eq…
Chvátal conjectured in 1973 the existence of some constant $t$ such that all $t$-tough graphs with at least three vertices are hamiltonian. While the conjecture has been proven for some special classes of graphs, it remains open in general. We say that a graph is $(K_2 \cup 3K_1)$-free if it contains no induced subgraph isomorphic to $K_2 \cup 3K_1$, where $K_2 \cup 3K_1$ is the disjoint union of an edge and three isolated vertices. In this paper, we show that every 3-tough $(K_2 \cup 3K_1)$-free graph with at least three vertices is hamiltonian.
Let $G$ be a $t$-tough graph on $n\ge 3$ vertices for some $t>0$. It was shown by Bauer et al. in 1995 that if the minimum degree of $G$ is greater than $\frac{n}{t+1}-1$, then $G$ is hamiltonian. In terms of Ore-type hamiltonicity conditions, the problem was only studied when $t$ is between 1 and 2. In this paper, we show that if the degree sum of any two nonadjacent vertices of $G$ is greater than $\frac{2n}{t+1}+t-2$, then $G$ is hamiltonian.
In this paper, we provide a method to find a Hamiltonian cycle in the prism of a 5-chordal graph, which is $(1+ε)$-tough, with some special conditions.
The vertex set of the kth cartesian power of a directed cycle of length m can be naturally identified with the set of k-tuples of integers modulo m. For any two vertices v and w of this graph, it is easy to see that if there is a hamiltonian path from v to w, then the sum of the coordinates of v is congruent, modulo m, to one more than the sum of the coordinates of w. We prove the converse, unless k = 2 and m is odd.
We show that if $C_1$ and $C_2$ are directed cycles (of length at least two), then the Cartesian product $C_1 \Box C_2$ has two arc-disjoint hamiltonian paths. (This answers a question asked by J. A. Gallian in 1985.) The same conclusion also holds for the Cartesian product of any four or more directed cycles (of length at least two), but some cases remain open for the Cartesian product of three directed cycles. We also discuss the existence of arc-disjoint hamiltonian paths in $2$-generated Cayley digraphs on (finite or infinite) abelian groups.
Let $G$ be a simple graph, and let $Δ(G)$ and $χ'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $χ'(G)=Δ(G)$ or $Δ(G)+1$. We say $G$ is $Δ$-critical if $χ'(G)=Δ+1$ and $χ'(H)<χ'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a $Δ$-critical graph, then $G$ has a 2-factor. Let $G$ be an $n$-vertex $Δ$-critical graph. It was proved that if $Δ(G)\ge n/2$, then $G$ has a 2-factor; and that if $Δ(G)\ge 2n/3+12$, then $G$ has a hamiltonian cycle, and thus a 2-factor. It is well known that every 2-tough graph with at le…
We show that an edge-dominating cycle in a $2K_2$-free graph can be found in polynomial time; this implies that every 1/(k-1)-tough $2K_2$-free graph admits a k-walk, and it can be found in polynomial time. For this class of graphs, this proves a long-standing conjecture due to Jackson and Wormald (1990). Furthermore, we prove that for any ε>0 every (1+ε)-tough $2K_2$-free graph is prism-Hamiltonian and give an effective construction of a Hamiltonian cycle in the corresponding prism, along with few other similar results.
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 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…
The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph on two vertices. A graph $G$ is prism-hamiltonian if the prism over $G$ is hamiltonian. We prove that every polyhedral graph (i.e. 3-connected planar graph) of minimum degree at least four is prism-hamiltonian.
For a 1-tough graph $G$ we define $σ_3(G) = \min\{°(u) + °(v)+ °(w):$ $\{u, v, w\}$ is an independent set of vertices$\}$ and $NC2(G)=\min \{|N(u)\cup N(v)|: d(u,v)=2\}$. D. Bauer, G. Fan and H.J.Veldman proved that $c(G)\geq \min\{n,2NC2(G)\}$ for any 1-tough graph $G$ with $σ_3(G)\geq n\geq 3$, where $c(G)$ is the circumference of $G$ (D. Bauer, G. Fan and H.J.Veldman,Hamiltonian properties of graphs with large neighborhood unions,Discrete Mathematics, 1991). They also conjectured a stronger upper bound for the circumference: $c(G)\geq\min\{n,2NC2(G)+4\}$.In this paper, we prove this conject…
The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. A graph $G$ is hamiltonian if there exists a spanning cycle in $G$, and $G$ is prism-hamiltonian if the prism over $G$ is hamiltonian. In [M.~Rosenfeld, D.~Barnette, Hamiltonian circuits in certain prisms, Discrete Math. 5 (1973), 389--394] the authors conjectured that every 3-connected planar graph is prism-hamiltonian. We construct a counterexample to the conjecture.
In this paper we study some open questions related to the smallest order $f({\cal C},\lnot {\cal H})$ of a 4-regular graph which has a connectivity property ${\cal C}$ but does not have a hamiltonian property ${\cal H}$. In particular, ${\cal C}$ is either connectivity, 2-connectivity or 1-toughness and ${\cal H}$ is hamiltonicity, homogeneously traceability or traceability. A standard theoretical approach to these questions had already been used in the literature, but did not succeed in determining the exact value of $f()$. Here we have chosen to use Integer Linear Programming and to encode t…
A toroidal grid graph is a Cartesian product of cycles, and the run length of a Hamiltonian cycle in a grid graph is defined to be the maximum number r such that any r consecutive edges include no more than one edge in any dimension. By constructive methods, we place bounds on the maximum run length possible for a Hamiltonian cycle in several families of grid graphs.
We study the list chromatic number of Cartesian products of graphs through the Alon-Tarsi number as defined by Jensen and Toft (1995) in their seminal book on graph coloring problems. The Alon-Tarsi number of $G$, $AT(G)$, is the smallest $k$ for which there is an orientation, $D$, of $G$ with max indegree $k\!-\!1$ such that the number of even and odd circulations contained in $D$ are different. It is known that $χ(G) \leq χ_\ell(G) \leq χ_p(G) \leq AT(G)$, where $χ(G)$ is the chromatic number, $χ_\ell(G)$ is the list chromatic number, and $χ_p(G)$ is the paint number of $G$. In this paper we…
The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. $G$ is prism-hamiltonian if the prism over $G$ has a Hamilton cycle. A good even cactus is a connected graph in which every block is either an edge or an even cycle, and every vertex is contained in at most two blocks. It is known that good even cacti are prism-hamiltonian. Indeed, showing the existence of a spanning good even cactus has become one of the most common techniques in proving prism-hamiltonicity. Špacapan asked whether having a spanning good even cactus is equivalent to having a hamiltonian p…
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…
A graph is an instrument which is extensively utilized to model various problems in different fields. Up to date, many graphs have been developed to represent algebraic structures, particularly rings in order to study their properties. In this article, by focusing on commutative ring $ R $, we introduce a new notion of unity product graph associated with $ R $ and its complement. In addition, we prove that if the number of vertices of the unity product graph is at least 2, then the graph is disconnected, while its complement graph is connected. Furthermore, it is shown that there are some comm…
