Petr Vrana, Xingzhi Zhan, Leilei Zhang
We prove that every 3-connected claw-free graph with domination number at most 3 is hamiltonian-connected. The result is sharp and it is inspired by a conjecture posed by Zheng, Broersma, Wang and Zhang in 2020.
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We prove that 3-connected triangulations with at most one separating triangle are hamiltonian-connected. In order to show bounds on the strongest form of this theorem, we proved that for any $s\geq4$ there are 3-connected triangulation with $s$ separating triangles that are not hamiltonian-connected. We also present computational results which show that all `small' 3-connected triangulations with at most 3 separating triangles are hamiltonian-connected.
4-regular 4-connected Hamiltonian graphs with few Hamiltonian cycles
该研究证明了存在哈密顿环数量有界的无穷多族4-正则4-连通哈密顿图，并提出5-正则5-连通情形下是否存在同类图族的开放问题。
The most general structure of graphs with hamiltonian or hamiltonian connected square
本文刻画使图平方 $G^2$ 为哈密顿图或哈密顿连通图的最一般块-割点结构，并借鉴哈密顿全图的研究思路。
4-connected 1-planar chordal graphs are Hamiltonian-connected
本文研究了4-连通1-平面弦图，利用1-平面4-树的特性作为证明工具，证明所有此类图均为哈密顿连通的。
We prove that every connected strongly regular graph on sufficiently many vertices is Hamiltonian. We prove this by showing that, apart from three families, connected strongly regular graphs are (highly) pseudo-random. Our results suggest a number of new questions and conjectures.
Tutte's $3$-flow conjecture says that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. Kochol (2001) showed that it is enough to prove this conjecture for $5$-edge-connected graphs. Former, Jaeger, Linial, Payan, and Tarsi (1992) conjectured that every $5$-edge-connected graph is $Z_3$-connected and so it admits a nowhere-zero $3$-flow. In this note, we show that if the second conjecture would be true, then every $3$-tree-connected graph must also be $Z_3$-connected and so Tutte's $3$-flow conjecture can be extended to this family of graphs.
Fullerene graphs, i.e., 3-connected planar cubic graphs with pentagonal and hexagonal faces, are conjectured to be Hamiltonian. This is a special case of a conjecture of Barnette and Goodey, stating that 3-connected planar graphs with faces of size at most 6 are Hamiltonian. We prove the conjecture.
We determine the possible maximum degrees of a minimally hamiltonian-connected graph with a given order. This answers a question posed by Modalleliyan and Omoomi in 2016. We also pose two unsolved problems.
Whitney proved in 1931 that every 4-connected planar triangulation is hamiltonian. Later in 1979, Hakimi, Schmeichel and Thomassen conjectured that every such triangulation on $n$ vertices has at least $2(n - 2)(n - 4)$ hamiltonian cycles. Along this direction, Brinkmann, Souffriau and Van Cleemput established a linear lower bound on the number of hamiltonian cycles in 4-connected planar triangulations. In stark contrast, Alahmadi, Aldred and Thomassen showed that every 5-connected triangulation of the plane or the projective plane has exponentially many hamiltonian cycles. This gives the moti…
Barnette conjectured that all cubic $3$-connected plane graphs with maximum face size at most $6$ are hamiltonian. We provide a method of construction of a hamiltonian cycle (in dual terms) in an arbitrary cubic, $3$-connected plane graph possessing such a face $g$ that every face incident with $g$ has at most $5$ edges and every other face has at most $6$ edges.
A graph $G$ is $l$-path Hamiltonian if every path of length not exceeding $l$ is contained in a Hamiltonian cycle. It is well known that a 2-connected, $k$-regular graph $G$ on at most $3k-1$ vertices is edge-Hamiltonian if for every edge $uv$ of $G$, $\{u,v\}$ is not a cut-set. Thus $G$ is 1-path Hamiltonian if $G\setminus \{u,v\}$ is connected for every edge $uv$ of $G$. Let $P=uvz$ be a 2-path of a 2-connected, $k$-regular graph $G$ on at most $2k$ vertices. In this paper, we show that there is a Hamiltonian cycle containing the 2-path $P$ if $G\setminus V(P)$ is connected. Therefore, the w…
Bouchet's conjecture for cyclically 5-edge-connected, cubic signed graphs
该研究证明了布切特猜想在循环五边连通立方符号图中成立，表明此类图具有处处非零的六流。
Hamiltonian cycles and Hamiltonian paths in $2k$-connected, $1$-tough and $(P_{3}\cup kP_{1})$-free graphs
本文针对k≥2，证明了2k-连通、1-坚韧且不含(P₃∪kP₁)子图的图必为哈密顿图，(2k+1)-连通、坚韧度大于1且不含(P₃∪kP₁)子图的图必为哈密顿连通图。
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, (…
Counterexamples to a conjecture of Hoa on maximal non-Hamiltonian graphs
针对1994年Hoa提出的极大非哈密顿图相关猜想，本文通过构造n≥56阶的反例，成功推翻了该猜想。
Thomassen conjectured that there is a function $f(k)$ such that every strongly $f(k)$-connected tournament contains $k$ edge-disjoint Hamiltonian cycles. This conjecture was recently proved by Kühn, Lapinskas, Osthus, and Patel who showed that $f(k)\leq O(k^2(\log k)^2)$ and conjectured that there is a constant $C$ such that $f(k)\leq Ck^2$. We prove this conjecture.
Hamiltonian Properties of 3-Connected Claw-Free Graphs and Line Graphs of 3-Hypergraphs
本文研究了3-连通无爪图的哈密顿性质，证明了支配数不超过5的此类图是哈密顿图，且支配数不超过4的图是哈密顿连通的，并探讨了3超图线图的哈密顿性。
Every 2-connected, cubic, planar graph with faces of size at most 6 is Hamiltonian
本文证明面大小至多为6的2-连通、三次、平面图均为哈密顿图且6-面条件紧致，从而将Barnette-Goodey猜想的连通性条件推至最弱可能。
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.
Komlós conjectured in 1981 that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when $d$ is sufficiently large. In fact we prove a stronger result: for large $d$, any graph $G$ with average degree at least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$, unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we specify.
C. Thomassen in \cite{[11]} suggested (see also \cite{[2]}, J. C.Bermond, C. Thomassen, Cycles in Digraphs - A survey, J. Graph Theory 5 (1981) 1-43, Conjectures 1.6.7 and 1.6.8) the following conjectures : 1. Every 3-strongly connected digraph of order $n$ and with minimum degree at least $n+1$ is strongly Hamiltonian-connected. 2. Let $D$ be a 4-strongly connected digraph of order $n$ such that the sum of the degrees of any pair of non-adjacent vertices is at least $2n+1$. Then $D$ is strongly Hamiltonian-connected. We disprove Conjecture 1 and prove two results which provide some support fo…
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.
On a Conjecture by Hayashi on Finite Connected Quandles
本文证明了Hayashi关于有限连通quandle轮廓中最长长度是其余长度倍数的猜想在轮廓长度不超过五时成立。
An extension of a cubic 2-connected plane graph G to a hamiltonian plane graph contained in G^{2}
该研究针对简单三次2-连通平面图，证明其含n个分支的2-因子对应存在包含于G²、满足边数与最大度条件的简单哈密顿平面图，拓展了相关图论结论。
Graphs with Many Hamiltonian Paths
本文构造了非哈密顿但几乎任意顶点对都有哈密顿路径的图族，并给出边数达到3n/2下界的无限哈密顿连通图族。
Rosenfeld in 1974 conjectured that there is an integer N > 8 such that every tournament of order n > N contains every non-directed cycle of order n. We prove that, with exactly 35 exceptions, every tournament of order n > 2 contains each non-directed cycle of order m, 2 < m < n+1.
Every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected
针对 Shi–Shan Hamilton 性猜想已获证实的背景，本文证明更强结论：韧度大于一的 $2k$-连通 $(P_2\cup kP_1)$-无关图必为 Hamilton 连通。
A Conjecture on Rainbow Hamiltonian Cycle Decomposition
本文完全解决了Wu于1999年提出的关于完全图$K_{2n+1}$中子图边分布于不同哈密顿圈的彩虹哈密顿圈分解猜想，并指出其在线性荫度猜想和受限尺寸Ramsey数中的应用。
Minimum degree ensuring that a hypergraph is hamiltonian-connected
本文研究 r-一致超图的哈密顿连通性，确定了所有 3≤r<n 情形下保证该性质的精确最小度下界，并揭示其与哈密顿 Berge 圈度界及普通图情形的差异。
Let $λ_{1}(G)$ and $μ_{1}(G)$ denote the spectral radius and the Laplacian spectral radius of a graph $G$, respectively. Li in [Electronic J. Linear Algebra 34 (2018) 389-392] proved sharp upper bounds of $λ_{1}(G)$ based on the connectivity to assure a connected graph to be Hamiltonian and traceable, respectively. In this paper, we present best possible upper bounds of $λ_{1}(G)$ for $k$-connected graphs to be Hamiltonian-connected and homogeneously traceable, respectively. Furthermore, best possible upper bounds of $μ_{1}(G)$ to predict $k$-connected graphs to be Hamiltonian-connected, Hamil…
Merker conjectured that if $k \ge 2$ is an integer and $G$ a 3-connected cubic planar graph of circumference at least $k$, then the set of cycle lengths of $G$ must contain at least one element of the interval $[k, 2k+2]$. We here prove that for every even integer $k \ge 6$ there is an infinite family of counterexamples.
Hasunuma [J. Graph Theory 102 (2023) 423-435] conjectured that for any tree $T$ of order $m$, every $k$-connected (or $k$-edge-connected) graph $G$ with minimum degree at least $k+m-1$ contains a tree $T'\cong T$ such that $G-E(T')$ is still $k$-connected (or $k$-edge connected). Hasunuma verified this conjecture for $k\leq 2$. In this paper, we confirm this conjecture for $k=3$.
Alphatrion conjectured that it is possible to label the vertices of an $n$-dimensional hypercube with distinct positive integers such that for every Hamiltonian path $a_1, \dots, a_{2^n},$ we have $a_i + a_{i+1}$ prime for all $i.$ We prove the conjecture by proving the more general result that a graph $G = (V, E)$ can be labeled with distinct positive integers such that the edge sum for all $e \in E$ is prime if and only if $G$ is bipartite.
In this paper we study the randomly edge colored graph that is obtained by adding randomly colored random edges to an arbitrary randomly edge colored dense graph. In particular we ask how many colors and how many random edges are needed so that the resultant graph contains a fixed number of edge disjoint rainbow Hamilton cycles. We also ask when in the resultant graph every pair of vertices is connected by a rainbow path.
A graph $G$ is Hamiltonian-connected if there exists a Hamiltonian path between any two vertices of $G$. It is known that if $G$ is 2-connected then the graph $G^2$ is Hamiltonian-connected. In this paper we prove that the square of every self-complementary graph of order grater than 4 is Hamiltonian-connected. If $G$ is a $k$-critical graph, then we prove that the Mycielski graph $μ(G)$ is $(k+1)$-critical graph. Jarnicki et al.[7] proved that for every Hamiltonian graph of odd order, the Mycielski graph $μ(G)$ of $G$ is Hamiltonian-connected. They also pose a conjecture that if $G$ is Hamilt…
Equivalent formulation of Thomassen's conjecture using Tutte paths in claw-free graphs
本文将Thomassen猜想与Jackson猜想的等价性进一步推广，证明在每个连通无爪图中任意两点都可由一条极大Tutte路径连接。
Say a graph $G$ is a {\em pentagraph} if every cycle has length at least five, and every induced cycle of odd length has length five. N. Robertson proposed the conjecture that the Petersen graph is the only pentagraph that is three-connected and internally 4-connected, but this was disproved by M. Plummer and X. Zha in 2014. Plummer and Zha conjectured that every 3-connected, internally 4-connected pentagraph is three-colourable. We prove this: indeed, we will prove that every pentagraph is three-colourable.
A conjecture of Thomassen from 1982 states that for every k there is an f(k) so that every strongly f(k)-connected tournament contains k edge-disjoint Hamilton cycles. A classical theorem of Camion, that every strongly connected tournament contains a Hamilton cycle, implies that f(1)=1. So far, even the existence of f(2) was open. In this paper, we prove Thomassen's conjecture by showing that f(k)=O(k^2*log^2(k)). This is best possible up to the logarithmic factor. As a tool, we show that every strongly 10^4*k*log(k)-connected tournament is k-linked (which improves a previous exponential bound…
Proof of Lovász conjecture for odd order
本文通过揭示奇数阶连通点传递图的结构，证明了Lovász猜想在奇数阶情形下成立，即所有奇数阶连通点传递图都是哈密顿图。
In 1972, Tutte posed the $3$-Flow Conjecture: that all $4$-edge-connected graphs have a nowhere zero $3$-flow. This was extended by Jaeger et al.(1992) to allow vertices to have a prescribed, possibly non-zero difference (modulo $3$) between the inflow and outflow. They conjectured that all $5$-edge-connected graphs with a valid prescription function have a nowhere zero $3$-flow meeting that prescription. Kochol (2001) showed that replacing $4$-edge-connected with $5$-edge-connected would suffice to prove the $3$-Flow Conjecture and Lovász et al.(2013) showed that both conjectures hold if the …
We say a graph $G$ has a Hamiltonian path if it has a path containing all vertices of $G$. For a graph $G$, let $σ_2(G)$ denote the minimum degree sum of two nonadjacent vertices of $G$; restrictions on $σ_2(G)$ are known as Ore-type conditions. Given an integer $t\geq 5$, we prove that if a connected graph $G$ on $n$ vertices satisfies $σ_2(G)>{t-3\over t-2}n$, then $G$ has either a Hamiltonian path or an induced subgraph isomorphic to $K_{1, t}$. Moreover, we characterize all $n$-vertex graphs $G$ where $σ_2(G)={t-3\over t-2}n$ and $G$ has neither a Hamiltonian path nor an induced subgraph i…
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…
There are many major open problems in integer flow theory, such as Tutte's 3-flow conjecture that every 4-edge-connected graph admits a nowhere-zero 3-flow, Jaeger et al.'s conjecture that every 5-edge-connected graph is $Z_3$-connected and Kochol's conjecture that every bridgeless graph with at most three 3-edge-cuts admits a nowhere-zero 3-flow (an equivalent version of 3-flow conjecture). Thomassen proved that every 8-edge-connected graph is $Z_3$-connected and therefore admits a nowhere-zero 3-flow. Furthermore, Lov$\acute{a}$sz, Thomassen, Wu and Zhang improved Thomassen's result to 6-edg…
A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at least 6 is hamiltonian. The result extends to claw-free graphs and to Hamilton-connectedness.
In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let $ω$ be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to cubic graphs with $ω\geq 2$. We show that if a cubic graph $G$ has no edge cut with fewer than $ {5/2} ω- 1$ edges that separates two odd cycles of a minimum 2-factor of $G$, then $G$ has a nowhere-zero 5-flow. This implies that if a cubic graph $G$ is cyclically $n$-edge connected and $n \geq {5/2} ω- 1$, then $G$ has a nowhere-zero 5-flow.
It is well-known that every planar 4-connected graph has a Hamiltonian cycle. In this paper, we study the question whether every 1-planar 4-connected graph has a Hamiltonian cycle. We show that this is false in general, even for 5-connected graphs, but true if the graph has a 1-planar drawing where every region is a triangle.
Given a graph $G$ and a graph property $P$ we say that $G$ is minimal with respect to $P$ if no proper induced subgraph of $G$ has the property $P$. An HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph $H$, a graph $G$ is $H$-free if $G$ has no induced subgraph isomorphic to $H$. The main motivation for this paper originates from a theorem of Duffus, Gould, and Jacobson (1981), which characterizes all the minimal connected graphs with no Hamiltonian path. In 1998, Brousek characterized all the claw-free HC-obstructions. On a similar note, Chiba and Furuya (2021), cha…
An Improved Interpolation Theorem and Disproofs of Two Conjectures on 2-Connected Subgraphs
本文改进了2-连通子图的插值定理，并驳斥了两个关于2-连通子图的猜想，同时提出了新的猜想。
Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices then $G$ contains at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar $n$-vertex triangulations $G$ that do not have too many separating 4-cyc…
Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we show that every $4$-connected planar triangulation on $n$ vertices has $Ω(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{Ω(n^{1/4})}$ Hamiltonian cycles.
On regular 2-path Hamiltonian graphs
本文研究2-路哈密顿图相关问题，刻画了说明2k界紧性的一类图，并证明排除该类图后，满足顶点数条件的2-连通、3-连通k-正则图在G\V(P)连通时存在含指定2-路的哈密顿圈。
In [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.], Mader conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $δ(G)\geq \lfloor\frac{3}{2}k\rfloor+m-1$ contains a subtree $T'$ isomorphic to $T$ such that $G-V(T')$ is $k$-connected. In the same paper, Mader proved that the conjecture is true when $T$ is a path. Diwan and Tholiya [A.A. Diwan, N.P. Tholiya, Non-separating trees in connected graphs, Discrete Math. 309 (2009) 5235-5237.] verified the conjecture when $k=1…
Mader [J. Graph Theory 65 (2010) 61-69] conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $δ(G)\geq \lfloor\frac{3}{2}k\rfloor+m-1$ contains a subtree $T'$ isomorphic to $T$ such that $G-V(T')$ is $k$-connected. The conjecture has been verified for paths, trees when $k=1$, and stars or double-stars when $k=2$. In this paper we verify the conjecture for two classes of trees when $k=2$. For digraphs, Mader [J. Graph Theory 69 (2012) 324-329] conjectured that every $k$-connected digraph $D$ with minimum semi-degree…
We disprove a conjecture of Frank stating that each weakly 2k-connected has a k-vertex-connected orientation. For k at least 3, we also prove that the problem of deciding whether a graph has a k-vertex-connected orientation is NP-complete.
The number of cycles of a given length in dense hamiltonian graphs: proving Hilton's conjecture
本文证明了Hilton猜想：对于阶数至少440的哈密顿图，若边数超过n^2/4+1，则对每个长度ℓ（3≤ℓ≤n），图中至少有n-ℓ+2个ℓ-圈。
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…
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…
Let $P_G(k)$ be the number of proper $k$-colorings of a finite simple graph $G$. Tomescu's conjecture, which was recently solved by Fox, He, and Manners, states that $P_G(k) \le k!(k-1)^{n-k}$ for all connected graphs $G$ on $n$ vertices with chromatic number $k\geq 4$. In this paper, we study the same problem with the additional constraint that $G$ is $\ell$-connected. For $2$-connected graphs $G$, we prove a tight bound \[ P_G(k) \le (k-1)!((k-1)^{n-k+1} + (-1)^{n-k}), \] and show that equality is only achieved if $G$ is a $k$-clique with an ear attached. For $\ell \ge 3$, we prove an asympt…
Proof a conjecture on connectivity keeping odd paths in k-connected bipartite graphs
本文证明 k-连通二部图在最小度至少 k+⌈(m+1)/2⌉ 时含一条删除后仍保持 k-连通的 m 阶路，从而证实奇路情形并改进偶路界。
There is empirical evidence supporting the claim that almost all cubic non-Hamiltonian graphs are bridge graphs. In this paper, we pose a related conjecture and prove that the original claim holds for non-3-connected graphs if the conjecture is true.
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