An upper bound of a generalized upper Hamiltonian number of a graph

Martin Dzúrik

Displaying similar documents to “An upper bound of a generalized upper Hamiltonian number of a graph”

Hamiltonian-colored powers of strong digraphs

Garry Johns, Ryan Jones, Kyle Kolasinski, Ping Zhang (2012)

Discussiones Mathematicae Graph Theory

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For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power $D^{k}$ of D is that digraph having vertex set V(D) with the property that (u, v) is an arc of $D^{k}$ if the directed distance $^{\to} d_{D} (u, v)$ from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph $D^{k}$ is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph $D^{k}$ is distance-colored if each arc (u, v) of $D^{k}$ is assigned the color i where $i =^{\to} d_{D} (u, v)$ . The digraph...

Spectral radius and Hamiltonicity of graphs with large minimum degree

Vladimir Nikiforov (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a graph of order $n$ and $λ (G)$ the spectral radius of its adjacency matrix. We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in $G$ . One of the main results of the paper is the following theorem: Let $k \geq 2,$ $n \geq k^{3} + k + 4,$ and let $G$ be a graph of order $n$ , with minimum degree $δ (G) \geq k .$ If $λ (G) \geq n - k - 1,$ then $G$ has a Hamiltonian cycle, unless $G = K_{1} \lor (K_{n - k - 1} + K_{k})$ or $G = K_{k} \lor (K_{n - 2 k} + {\bar{K}}_{k}) .$

Hamiltonicity of cubic Cayley graphs

Henry Glover, Dragan Marušič (2007)

Journal of the European Mathematical Society

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Following a problem posed by Lovász in 1969, it is believed that every finite connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from finite groups having a $(2, s, 3)$ -presentation, that is, for groups $G = 〈 a, b ∣ a^{2} = 1, b^{s} = 1, {(a b)}^{3} = 1, \dots 〉$ generated by an involution $a$ and an element $b$ of order $s \geq 3$ such that their product $a b$ 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...

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Marcel Guardia, Vadim Kaloshin (2015)

Journal of the European Mathematical Society

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We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix $s > 1$ . Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with $s$ -Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is $c > 0$ such that for any $𝒦 ≫ 1$ we find a solution $u$ and a time $T$ such that ${∥ u (T) ∥}_{H^{s}} \geq 𝒦 {∥ u (0) ∥}_{H^{s}}$ . Moreover, the time $T$ satisfies the polynomial bound $0 < T < 𝒦^{C}$ .

Periodic solutions for a class of non-autonomous Hamiltonian systems with $p (t)$ -Laplacian

Zhiyong Wang, Zhengya Qian (2024)

Mathematica Bohemica

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We investigate the existence of infinitely many periodic solutions for the $p (t)$ -Laplacian Hamiltonian systems. By virtue of several auxiliary functions, we obtain a series of new super- $p^{+}$ growth and asymptotic- $p^{+}$ growth conditions. Using the minimax methods in critical point theory, some multiplicity theorems are established, which unify and generalize some known results in the literature. Meanwhile, we also present an example to illustrate our main results are new even in the case $p (t) \equiv p = 2$ . ...

Degree sums of adjacent vertices for traceability of claw-free graphs

Tao Tian, Liming Xiong, Zhi-Hong Chen, Shipeng Wang (2022)

Czechoslovak Mathematical Journal

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The line graph of a graph $G$ , denoted by $L (G)$ , has $E (G)$ as its vertex set, where two vertices in $L (G)$ are adjacent if and only if the corresponding edges in $G$ have a vertex in common. For a graph $H$ , define ${\bar{σ}}_{2} (H) = min {d (u) + d (v) : u v \in E (H)}$ . Let $H$ be a 2-connected claw-free simple graph of order $n$ with $δ (H) \geq 3$ . We show that, if ${\bar{σ}}_{2} (H) \geq \frac{1}{7} (2 n - 5)$ and $n$ is sufficiently large, then either $H$ is traceable or the Ryjáček’s closure $cl (H) = L (G)$ , where $G$ is an essentially $2$ -edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10...

On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. III. Strengthening of the characterizations performed in parts I and II, for Lagrangian systems. An invariance property.

Aldo Bressan (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In [1] I and II various equivalence theorems are proved; e.g. an ODE $(\mathcal{E})\dot{z}=F(t,z,u,\dot{u})\,(\in\mathbb{R}^{m})$ with a scalar control $u=u(\cdot)$ is linear w.r.t. $\dot{u}$ iff $(\alpha)$ its solution $z(u,\cdot)$ with given initial conditions (chosen arbitrarily) is continuous w.r.t. $u$ in a certain sense, or iff $(\beta)$ $z(u,\cdot)$ satisfies certain conditions by which $1^{st}$ -order discontinuities of $u$ and $\dot{u}$ can be treated satisfactorily. In the case when, for $z=(q,p)$ equation $(\mathcal{E})$ is a semi-Hamiltonian system, equivalent to a system of Lagrangian equations of a general type,...

The Turán number of the graph $3 P_{4}$

Halina Bielak, Sebastian Kieliszek (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $e x (n, G)$ denote the maximum number of edges in a graph on $n$ vertices which does not contain $G$ as a subgraph. Let $P_{i}$ denote a path consisting of $i$ vertices and let $m P_{i}$ denote $m$ disjoint copies of $P_{i}$ . In this paper we count $e x (n, 3 P_{4})$ .

Note on improper coloring of $1$ -planar graphs

Yanan Chu, Lei Sun, Jun Yue (2019)

Czechoslovak Mathematical Journal

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A graph $G = (V, E)$ is called improperly $(d_{1}, \dots, d_{k})$ -colorable if the vertex set $V$ can be partitioned into subsets $V_{1}, \dots, V_{k}$ such that the graph $G [V_{i}]$ induced by the vertices of $V_{i}$ has maximum degree at most $d_{i}$ for all $1 \leq i \leq k$ . In this paper, we mainly study the improper coloring of $1$ -planar graphs and show that $1$ -planar graphs with girth at least $7$ are $(2, 0, 0, 0)$ -colorable.

Saturation numbers for linear forests $P_{6} + t P_{2}$

Jingru Yan (2023)

Czechoslovak Mathematical Journal

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A graph $G$ is $H$ -saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$ . The saturation number, $sat (n, H)$ , is the minimum number of edges of a graph in the set of all $H$ -saturated graphs of order $n$ . We determine the saturation number $sat (n, P_{6} + t P_{2})$ for $n \geq \frac{10}{3} t + 10$ and characterize the extremal graphs for $n > \frac{10}{3} t + 20$ .

On the multiplicity of Laplacian eigenvalues for unicyclic graphs

Fei Wen, Qiongxiang Huang (2022)

Czechoslovak Mathematical Journal

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Let $G$ be a connected graph of order $n$ and $U$ a unicyclic graph with the same order. We firstly give a sharp bound for $m_{G} (μ)$ , the multiplicity of a Laplacian eigenvalue $μ$ of $G$ . As a straightforward result, $m_{U} (1) \leq n - 2$ . We then provide two graph operations (i.e., grafting and shifting) on graph $G$ for which the value of $m_{G} (1)$ is nondecreasing. As applications, we get the distribution of $m_{U} (1)$ for unicyclic graphs on $n$ vertices. Moreover, for the two largest possible values of $m_{U} (1) \in {n - 5, n - 3}$ , the corresponding graphs $U$ are...

Even factor of bridgeless graphs containing two specified edges

Nastaran Haghparast, Dariush Kiani (2018)

Czechoslovak Mathematical Journal

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An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Let $G$ be a bridgeless simple graph with minimum degree at least $3$ . Jackson and Yoshimoto (2007) showed that $G$ has an even factor containing two arbitrary prescribed edges. They also proved that $G$ has an even factor in which each component has order at least four. Moreover, Xiong, Lu and Han (2009) showed that for each pair of edges $e_{1}$ and $e_{2}$ of $G$ , there is an even factor containing $e_{1}$ and $e_{2}$ ...

Some results on the co-intersection graph of submodules of a module

Lotf Ali Mahdavi, Yahya Talebi (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a ring with identity and $M$ be a unitary left $R$ -module. The co-intersection graph of proper submodules of $M$ , denoted by $Ω (M)$ , is an undirected simple graph whose vertex set $V (Ω)$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N + K \neq M$ . We study the connectivity, the core and the clique number of $Ω (M)$ . Also, we provide some conditions on the module $M$ , under which the clique number of $Ω (M)$ is infinite and $Ω (M)$ is a planar graph. Moreover, we give...

Derivation of Hartree’s theory for mean-field Bose gases

Mathieu Lewin (2013)

Journées Équations aux dérivées partielles

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This article is a review of recent results with Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty and Jan Philip Solovej. We consider a system of $N$ bosons with an interaction of intensity $1 / N$ (mean-field regime). In the limit $N \to \infty$ , we prove that the first order in the expansion of the eigenvalues of the many-particle Hamiltonian is given by the nonlinear Hartree theory, whereas the next order is predicted by the Bogoliubov Hamiltonian. We also discuss the occurrence of Bose-Einstein condensation...

On the diameter of the intersection graph of a finite simple group

Xuanlong Ma (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group. The intersection graph $Δ_{G}$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of $G$ , and two distinct vertices $X$ and $Y$ are adjacent if $X \cap Y \neq 1$ , where $1$ denotes the trivial subgroup of order $1$ . A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters...