Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2024
Volume
Vol. 44
Number
No. 6
Description
Journal Volume
Opuscula Mathematica
Vol. 44 (2024)
Projects
Pages
Articles
Closure results for arbitrarily partitionable graphs
(Wydawnictwa AGH, 2024) Bensmail, Julien
A well-known result of Bondy and Chvátal establishes that a graph of order $n$ is Hamiltonian if and only if its $n$-closure (obtained through repeatedly adding an edge joining any two non-adjacent vertices with degree sum at least $n$) also is. In this work, we investigate such closure results for arbitrarily partitionable graphs, a weakening of Hamiltonian graphs being those graphs that can be partitioned into arbitrarily many connected graphs of arbitrary orders. Among other results, we establish closure results for arbitrary partitions into connected graphs of order at most 3, for arbitrary partitions into connected graphs of order exactly any $\lambda$, and for the property of being arbitrarily partitionable in full.
On a nonlocal p(x)-Laplacian Dirichlet problem involving several critical Sobolev-Hardy exponents
(Wydawnictwa AGH, 2024) Costa, Augusto César dos Reis; da Silva, Ronaldo Lopes
The aim of this work is to present a result of multiplicity of solutions, in generalized Sobolev spaces, for a non-local elliptic problem with $p(x)$-Laplace operator containing $k$ distinct critical Sobolev-Hardy exponents combined with singularity points $\textcolor{white}\$ \begin{cases} M(\psi(u)) (- \Delta_{p(x)} u + |u|^{p(x)-2} u) = \sum_{i=1}^{k} h_i(x) \dfrac{|u|^{p^*_{s_i}(x)-2} u}{|x|^{s_i(x)}} + f(x,u) & \text{ in }\Omega, \\ u=0 & \text{ on }\partial \Omega, \end{cases} \textcolor{white}\$$ where $\Omega\subset \mathbb{R}^N$ is a bounded domain with $0 \in \Omega$ and $1 \lt p^- \leq p(x) \leq p^+ \lt N$. The real function $M$ is bounded in $[0, +\infty)$ and the functions $h_i$ $(i=1, \ldots, k)$ and $f$ are functions whose properties will be given later. To obtain the result we use the Lions' concentration-compactness principle for critical Sobolev-Hardy exponent in the space $W^{1,p(x)}_{0}(\Omega)$ due to Yu, Fu and Li, and the Fountain Theorem.
Facial graceful coloring of plane graphs
(Wydawnictwa AGH, 2024) Czap, Július
Let $G$ be a plane graph. Two edges of $G$ are facially adjacent if they are consecutive on the boundary walk of a face of $G$. A facial edge coloring of $G$ is an edge coloring such that any two facially adjacent edges receive different colors. A facial graceful $k$-coloring of $G$ is a proper vertex coloring $c:V(G)\rightarrow\{1,2,\dots,k\}$ such that the induced edge coloring $c^{\prime}:E(G)\rightarrow\{1,2,\dots,k-1\}$ defined by $c^{\prime(uv)}=|c(u)-c(v)|$ is a facial edge coloring. The minimum integer $k$ for which $G$ has a facial graceful $k$-coloring is denoted by $\chi_{fg}(G)$. In this paper we prove that $\chi_{fg}(G)\leq 14$ for every plane graph $G$ and $\chi_{fg}(H)\leq 9$ for every outerplane graph $H$. Moreover, we give exact bounds for cacti and trees.
Positive solutions of nonpositone sublinear elliptic problems
(Wydawnictwa AGH, 2024) Godoy, Tomas
Consider the problem $-\Delta u=\lambda f(\cdot, u) $ in $\Omega$, $u=0$ on $\partial\Omega$, $u\gt 0$ in $\Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $C^{2}$ boundary when $n\geq2$, $\lambda\gt 0$, and where $f\in C (\overline{\Omega}\times[0,\infty)) $ satisfies $\lim_{s\rightarrow\infty}s^{-p}f(\cdot, s) =\gamma$ for some $p\in(0,1)$ and some $\gamma\in C(\overline{\Omega}) $ such that $\gamma\neq 0$ a.e. in $\Omega$ and, for some positive constants $c$ and $c^{\prime}$, $\gamma^{-}\leq cd_{\Omega}^{\beta}$ for some $\beta\in (\frac{n-1}{n},\infty)$ and $(-\Delta)^{-1}\gamma\geq c^{\prime}d_{\Omega}$, where $d_{\Omega}(x):=dist ( x,\partial \Omega) $ and $\gamma^{-}:=-\min(0,\gamma)$. Under these assumptions we show that for $\lambda$ large enough, the above problem has a positive weak solution $u\in C^{1}(\overline{\Omega})$ such that, for some constant $c^{\prime\prime}\gt 0$, $u\geq c^{\prime\prime}d_{\Omega}$ in $\Omega$.
Symmetric 2×2 matrix functions with order preserving property
(Wydawnictwa AGH, 2024) Štoudková Růžičková, Viera
It is known that the discrete matrix Riccati equation has the order preserving property under some assumptions. In this paper we formulate and prove the converse statement for the case when the dimensions of the matrices are $2 \times 2$ and the order preserving property holds for all such symmetric matrices