For a graph $G$, the $t$-th power $G^t$ is the graph on $V(G)$ such that two vertices are adjacent if and only if they have distance at most $t$ in $G$; and the $t$-th bi-power $G_B^t$ is the graph on $V(G)$ such that two vertices are adjacent if and only if their distance in $G$ is odd at most $t$. Fleischner's theorem states that the square of every 2-connected finite graph has a Hamiltonian cycle. Georgakopoulos prove that the square of every 2-connected infinite locally finite graph has a Hamiltonian circle. In this paper, we consider the Hamiltonicity of the bi-power of bipartite graphs. We show that for every connected finite bipartite graph $G$ with a perfect matching, $G_B^3$ has a Hamiltonian cycle. We also show that if $G$ is a connected infinite locally finite bipartite graph with a perfect matching, then $G_B^3$ has a Hamiltonian circle.
We prove several new sufficient conditions for hamiltonicity and bipancyclicity in balanced bipartite digraphs, in terms of sums of degrees over dominating or dominated pairs of vertices.
We prove a sharp Meyniel-type criterion for hamiltonicity of a balanced bipartite digraph: For k greater than or equal to 2, a bipartite digraph D with colour classes of cardinalities k is hamiltonian if the sum of degrees of vertices u and v is at least 3k+1 for every pair of vertices u, v such that D does not contain the arc uv nor vu. As a consequence, we obtain a sharp sufficient condition for hamiltonicity in terms of the minimal degree: a balanced bipartite digraph D on 2k vertices is hamiltonian if its minimal degree is at least (3k + 1)/2.
We prove that a strongly connected balanced bipartite digraph $D$ of order $2a$ is hamiltonian, provided $a\geq3$ and $d(x)+d(y)\geq 3a$ for every pair of vertices $x$, $y$ with a common in-neighbour or a common out-neighbour in $D$.
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 $y\in V(D…
In this paper, we discuss the Hamiltonicity of graphs in terms of Wiener index, hyper-Wiener index and Harary index of their quasi-complement or complement. Firstly, we give some sufficient conditions for an balanced bipartite graph with given the minimum degree to be traceable and Hamiltonian, respectively. Secondly, we present some sufficient conditions for a nearly balanced bipartite graph with given the minimum degree to be traceable. Thirdly, we establish some conditions for a graph with given the minimum degree to be traceable and Hamiltonian, respectively. Finally, we provide some condi…
In this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result for balanced bipartite graphs. Additionally, we construct infinitely many graphs to show that results proved in this paper give new strength for one to determine the Hamiltonicity of graphs.
A bipartite graph is chordal bipartite if every cycle of length at least six has a chord in it. M$\ddot{\rm u}$ller \cite {muller1996Hamiltonian} has shown that the Hamiltonian cycle problem is NP-complete on chordal bipartite graphs by presenting a polynomial-time reduction from the satisfiability problem. The microscopic view of the reduction instances reveals that the instances are $P_9$-free chordal bipartite graphs, and hence the status of Hamiltonicity in $P_8$-free chordal bipartite graphs is open. In this paper, we identify the first non-trivial subclass of $P_8$-free chordal bipartite…
We prove that for all countable tournaments $D$ the recently discovered compactification $|D|$ by their ends and limit edges contains a topological Hamilton path: a topological arc that contains every vertex. If $D$ is strongly connected, then $|D|$ contains a topological Hamilton circle. These results extend well-known theorems about finite tournaments, which we show do not extend to the infinite in a purely combinatorial setting.
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.
In this paper, we prove tight sufficient conditions for traceability and Hamiltonicity of connected graphs with given minimum degree, in terms of Wiener index and Harary index. We also prove some result on Hamiltonicity of balanced bipartite graphs in the similar fashion. In two recent papers \cite{LDJ2016,LDJ2017}, Liu et al. corrected some previous work on traceability of connected graphs in terms of Wiener index and Harary index, respectively, such as \cite{HW2013,Y2013}. We generalize these results and give short and unified proofs. All results in this paper are best possible.
For a connected graph, the Hamiltonian cycle (path) is a simple cycle (path) that spans all the vertices in the graph. It is known from \cite{muller,garey} that HAMILTONIAN CYCLE (PATH) are NP-complete in general graphs and chordal bipartite graphs. A convex bipartite graph $G$ with bipartition $(X,Y)$ and an ordering $X=(x_1,\ldots,x_n)$, is a bipartite graph such that for each $y \in Y$, the neighborhood of $y$ in $X$ appears consecutively. $G$ is said to have convexity with respect to $X$. Further, convex bipartite graphs are a subclass of chordal bipartite graphs. In this paper, we present…
Using topological circles in the Freudenthal compactification of a graph as infinite cycles, we extend to locally finite graphs a result of Oberly and Sumner on the Hamiltonicity of finite graphs. This answers a question of Stein, and gives a sufficient condition for Hamiltonicity in locally finite graphs.
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…
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.
Among the well-known sufficient degree conditions for the Hamiltonicity of a finite graph, the condition of Asratian and Khachatrian is the weakest and thus gives the strongest result. Diestel conjectured that it should extend to locally finite infinite graphs~$G$, in that the same condition implies that the Freudenthal compactification of $G$ contains a circle through all its vertices and ends. We prove Diestel's conjecture for claw-free graphs.
It is a well-known fact that hamiltonicity in planar cubic graphs is an NP-complete problem. This implies that the existence of an A-trail in plane eulerian graphs is also an NP-complete problem even if restricted to planar 3-connected eulerian graphs. In this paper we deal with hamiltonicity in planar cubic graphs G having a facial 2-factor Q via (quasi) spanning trees of faces in G/Q and study the algorithmic complexity of finding such (quasi) spanning trees of faces. We show, in particular, that if Barnette's Conjecture is false, then hamiltonicity in 3-connected planar cubic bipartite grap…
Tutte showed that $4$-connected planar graphs are Hamiltonian, but it is well known that $3$-connected planar graphs need not be Hamiltonian. We show that $K_{2,5}$-minor-free $3$-connected planar graphs are Hamiltonian. This does not extend to $K_{2,5}$-minor-free $3$-connected graphs in general, as shown by the Petersen graph, and does not extend to $K_{2,6}$-minor-free $3$-connected planar graphs, as we show by an infinite family of examples.
We study the existence of hamiltonian cycles in plane cubic graphs G having a facial 2-factor Q. Thus hamiltonicity in G is transformed into the existence of a (quasi) spanning tree of faces in the contraction G/Q. In particular, we study the case where G is the leapfrog extension (called vertex envelope in (Discrete Math., 309(14):4793-4809, 2009)) of a plane cubic graph G_0. As a consequence we prove hamiltonicity in the leapfrog extension of planar cubic cyclically 4-edge-connected bipartite graphs. This and other results of this paper establish partial solutions of Barnette's Conjecture ac…
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.
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.
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…
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 well-known twenty types of 2-uniform tilings of the plane give rise infinitely many doubly semi-equivelar maps on the torus. In this article, we show that every such doubly semi-equivelar map on the torus contains a Hamiltonian cycle. As a consequence, we establish the Nash-Williams conjecture for the graphs associated with these doubly semi-equivelar maps by showing that these graphs are either 3-connected or 4-connected.
We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs. We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing $K^4$ or $K_{2,3}$ as a minor is Hamiltonian if and only if it is $2$-connected. We show furthermore that, if a Hamilton circle exists in such a graph, then it is unique and spanned by the $2$-contractible edges. The third result of this paper is a construction of a graph which answers …
For locally finite infinite graphs the notion of Hamilton cycles can be extended to Hamilton circles, homeomorphic images of $S^1$ in the Freudenthal compactification. In this paper we extend some well-known theorems of the Hamiltonicity of finite Cayley graphs to infinite Cayley graphs.
We conjecture that a 2-connected graph $G$ of order $n$, in which $d(x)+d(y)\geq n-k$ for every pair of non-adjacent vertices $x$ and $y$, contains a cycle of length $n-k$ ($k<n/2$), unless $G$ is bipartite and $n-k$ is odd. This generalizes to long cycles a well-known degree sum condition for hamiltonicity of Ore. The conjecture is shown to hold for $k=1$.
In this paper, the concept of cyclic subsets in graph theory is introduced. An interesting theorem which relates to the collective Hamiltonicity of these cyclic subsets in graphs is also presented. This paper uses this theorem to construct an inductive proof of Barnette's long-standing conjecture, which asks whether every cubic, polyhedral, bipartite graph is Hamiltonian. Finding a class of graphs that are certain to be Hamiltonian is one of the biggest unsolved problems in Hamiltonian graph theory today.
We prove a refinement of the tree packing theorem by Tutte/Nash-Williams for finite graphs. This result is used to obtain a similar result for end faithful spanning tree packings in certain infinite graphs and consequently to establish a sufficient Hamiltonicity condition for the line graphs of such graphs.
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 (con…
We consider an infinite version of the bipartite Turán problem. Let $G$ be an infinite graph with $V(G) = \mathbb{N}$ and let $G_n$ be the $n$-vertex subgraph of $G$ induced by the vertices $\{1,2, \dots, n \}$. We show that if $G$ is $K_{2,t+1}$-free then for infinitely many $n$, $e(G_n) \leq 0.471 \sqrt{t} n^{3/2}$. Using the $K_{2,t+1}$-free graphs constructed by Füredi, we construct an infinite $K_{2,t+1}$-free graph with $e(G_n) \geq 0.23 \sqrt{t}n^{3/2}$ for all $n \geq n_0$.
The bipartite independence number of a graph $G$, denoted as $\tildeα(G)$, is the minimal number $k$ such that there exist positive integers $a$ and $b$ with $a+b=k+1$ with the property that for any two sets $A,B\subseteq V(G)$ with $|A|=a$ and $|B|=b$, there is an edge between $A$ and $B$. McDiarmid and Yolov showed that if $δ(G)\geq\tilde α(G)$ then $G$ is Hamiltonian, extending the famous theorem of Dirac which states that if $δ(G)\geq |G|/2$ then $G$ is Hamiltonian. In 1973, Bondy showed that, unless $G$ is a complete bipartite graph, Dirac's Hamiltonicity condition also implies pancyclici…
Let $G$ be a graph of even order and let $K_{G}$ be the complete graph on the same vertex set of $G$. A pairing of a graph $G$ is a perfect matching of the graph $K_{G}$. A graph $G$ has the Pairing-Hamiltonian property (for short, the PH-property) if for each one of its pairings, there exists a perfect matching of $G$ such that the union of the two gives rise to a Hamiltonian cycle of $K_G$. In 2015, Alahmadi \emph{et al.} gave a complete characterisation of the cubic graphs having the PH-property. Most naturally, the next step is to characterise the quartic graphs that have the PH-property. …
In 1978, Richard Rado showed that every edge-coloured complete graph of countably infinite order can be partitioned into monochromatic paths of different colours. He asked whether this remains true for uncountable complete graphs and a notion of \emph{generalised paths}. In 2016, Daniel Soukup answered this in the affirmative and conjectured that a similar result should hold for complete bipartite graphs with bipartition classes of the same infinite cardinality, namely that every such graph edge-coloured with $r$ colours can be partitioned into $2r-1$ monochromatic generalised paths with each …
Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian. A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching. Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces. We show a similar reduc…
A graph $G$ has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of $G$ such that the union of the two perfect matchings yields a Hamiltonian cycle of $G$. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and Häggkvist, combines three well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we study these concepts for cubic graphs in an attempt to characterise those cubic graphs for which every perfect matching corresponds to one …
The upper density of an infinite graph $G$ with $V(G) \subseteq \mathbb{N}$ is defined as $\overline{d}(G) = \limsup_{n \rightarrow \infty}{|V(G) \cap \{1,\ldots,n\}|}/{n}$. Let $K_{\mathbb{N}}$ be the infinite complete graph with vertex set $\mathbb{N}$. Corsten, DeBiasio, Lamaison and Lang showed that in every $2$-edge-colouring of $K_{\mathbb{N}}$, there exists a monochromatic path with upper density at least $(12 + \sqrt{8})/17$, which is best possible. In this paper, we extend this result to $k$-edge-colouring of $K_{\mathbb{N}}$ for $k \ge 3$. We conjecture that every $k$-edge-coloured $…
We prove that there exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than 2. We do this by proving a variant of a conjecture of Bilu and Linial about the existence of good 2-lifts of every graph. We also establish the existence of infinite families of `irregular Ramanujan' graphs, whose eigenvalues are bounded by the spectral radius of their universal cover. Such families were conjectured to exist by Linial and others. In particular, we prove the existence of infinite families of (c,d)-biregular bipartite graphs with all non-trivial eigenvalues bounded by sq…
Bipartite Graphs Are Not Well-Quasi-Ordered by Bipartite Minors
David R. Cheriton School of Computer Science, University of Waterloo（滑铁卢大学 David R. Cheriton 计算机科学学院）；Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院应用数学系）
We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette, Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and on even graphs in digraphs whose contraction is strongly connected (Hochstättler). We show that all of them fit into the same framework related to cuts in matchings. This allows us to find a counterexample to the conjecture of Hochstättler and show that the conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices. Finally, we state a new conjecture on bipartite cubic oriented graphs, that naturally arises in this setting.
