Yang et al. proved that every 3-connected, essentially 11-connected line graph is Hamilton-connected. This was extended by Li and Yang to 3-connected, essentially 10-connected graphs. Strengthening their result further, we prove that 3-connected, essentially 9-connected line graphs are Hamilton-connected. We use a method based on quasigraphs in combination with the discharging technique. The result extends to claw-free graphs.
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.
In 1980, Jackson proved that every 2-connected $k$-regular graph with at most $3k$ vertices is Hamiltonian. This result has been extended in several papers. In this note, we determine the minimum number of vertices in a connected $k$-regular graph that is not Hamiltonian, and we also solve the analogous problem for Hamiltonian paths. Further, we characterize the smallest connected $k$-regular graphs without a Hamiltonian cycle.
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.
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.
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…
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.
The connected graph of degree sequence 3,3,3,1,1,1 is called a net, and the vertices of degree 1 in a net is called its endvertices. Broersma conjectured in 1993 that a 2-connected graph G with no induced K_{1,3} is hamiltonian if every endvertex of each induced net of G has degree at least (|V(G)|-2)/3. In this paper we prove this conjecture in the affirmative.
Broersma and Veldman proved that every 2-connected claw-free and $P_6$-free graph is hamiltonian. Chen et al. extended this result by proving every 2-connected claw-heavy and $P_6$-free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and $P_6$-\emph{o}-heavy but not hamiltonian. In this paper we further give some Ore-type degree conditions restricting to induced $P_6$'s of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.
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…
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. …
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…
It is known that if G is a connected simple graph, then G^3 is Hamiltonian (in fact, Hamilton-connected). A simple graph is k-ordered Hamiltonian if for any sequence v_1, v_2, ..., v_k of k vertices there is a Hamiltonian cycle containing these vertices in the given order. In this paper, we prove that G^(3k/2 + 1) is k-ordered Hamiltonian for a connected graph G on at least k vertices. We further show that if G is connected, then G^4 is 4-ordered Hamiltonian and that if G is Hamiltonian, then G^3 is 5-ordered Hamiltonian. We also give bounds on the smallest power p_k such that G^p_k is k-order…
Let $G$ be a $k$ - connected ($k \geq 2$) graph of order $n$. If $χ(G) \geq n - k$, then $G$ is Hamiltonian or $K_k \vee (K_k^c \cup K_{n - 2k})$ with $n \geq 2 k + 1$, where $χ(G)$ is the chromatic number of the graph $G$.
A well-known conjecture of Grünbaum and Nash-Williams proposes that 4-connected toroidal graphs are hamiltonian. The corresponding results for 4-connected planar and projective-planar graphs were proved by Tutte and by Thomas and Yu, respectively, using induction arguments that proved a stronger result, that every edge is on a hamilton cycle. However, this stronger property does not hold for 4-connected toroidal graphs: Thomassen constructed counterexamples. Thus, the standard inductive approach will not work for the torus. One possible way to modify it is by characterizing the situations wher…
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 $D$.
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.
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…
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 $G$ 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…
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-…
In [Graphs Combin.~24 (2008) 469--483.], the third author and the fifth author conjectured that if $G$ is a $k$-connected graph such that $σ_{k+1}(G) \ge |V(G)|+κ(G)+(k-2)(α(G)-1)$, then $G$ contains a Hamiltonian cycle, where $σ_{k+1}(G)$, $κ(G)$ and $α(G)$ are the minimum degree sum of $k+1$ independent vertices, the connectivity and the independence number of $G$, respectively. In this paper, we settle this conjecture. This is an improvement of the result obtained by Li: If $G$ is a $k$-connected graph such that $σ_{k+1}(G) \ge |V(G)|+(k-1)(α(G)-1)$, then $G$ is Hamiltonian. The degree sum …
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.
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 …
In 1962, Erdős proved that if a graph $G$ with $n$ vertices satisfies $$ e(G)>\max\left\{\binom{n-k}{2}+k^2,\binom{\lceil(n+1)/2\rceil}{2}+\left\lfloor \frac{n-1}{2}\right\rfloor^2\right\}, $$ where the minimum degree $δ(G)\geq k$ and $1\leq k\leq(n-1)/2$, then it is Hamiltonian. For $n \geq 2k+1$, let $E^k_n=K_{k}\vee (kK_1+K_{n-2k})$, where "$\vee$" is the "join" operation. One can observe $e(E^k_n)=\binom{n-k}{2}+k^2$ and $E^k_n$ is not Hamiltonian. As $E^k_n$ contains induced claws for $k\geq 2$, a natural question is to characterize all 2-connected claw-free non-Hamiltonian graphs with th…
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…
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.
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…
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…
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\…
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 …
We analyse a polyhedron which contains the convex hull of all Hamiltonian cycles of a given undirected connected cubic graph. Our constructed polyhedron is defined by polynomially-many linear constraints in polynomially-many continuous (relaxed) variables. Clearly, the emptiness of the constructed polyhedron implies that the graph is non-Hamiltonian. However, whenever a constructed polyhedron is non-empty, the result is inconclusive. Hence, the following natural question arises: if we assume that a non-empty polyhedron implies Hamiltonicity, how frequently is this diagnosis incorrect? We prove…
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 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…
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.
Tight spectral conditions for the Hamiltonicity of $K_{1,r}$-free split graphs
School of Mathematical Sciences, South China Normal University（华南师范大学数学科学学院）；School of Computer Science, South China Normal University（华南师范大学计算机学院）；School of Mathematics and Systems Science, Guangdong Polytechnic Normal University（广东技术师范大学数学与系统科学学院）；Department of Mathematics, West Virginia University（西弗吉尼亚大学数学系）
本文研究了 $K_{1,r}$-free split 图的 Hamilton 性，针对 $r=3,4$ 情况，给出紧谱条件以确保图的 Hamilton 性。
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.
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…
In this paper we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on n labeled vertices. At each round we are presented with K=K(n) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the current graph. The goal is to create a Hamilton cycle as soon as possible. We show that this problem has three regimes, depending on the value of K. For K=o(\log n), the threshold for Hamiltonicity is (1+o(1))n\log n /(2K), i.e., typically we can construct a Hamilton cycle K times faster that in the usua…
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 …
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…
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…
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…
On the hamiltonicity problem of bicirculants: a reduction to cyclic Haar graphs
Università di Modena e Reggio Emilia（摩德纳和雷焦艾米利亚大学）；University of Primorska（滨海大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；University of Ljubljana（卢布尔雅那大学）
