Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2025
Volume
Vol. 45
Number
No. 6
Description
Journal Volume
Opuscula Mathematica
Vol. 45 (2025)
Projects
Pages
Articles
A note on distance labeling of graphs
(Wydawnictwa AGH, 2025) Casselgren, Carl Johan; Henricsson, Anders
We study labelings of connected graphs $G$ using labels $1,\ldots,|V(G)|$ encoding the distances between vertices in $G$. Following Lennerstad and Eriksson [Electron. J. Graph Theory Appl. 6 (2018), 152-165], we say that a graph $G$ which has a labeling $c$ satisfying that $d(u,v) \lt d(x,y) \Rightarrow c(u,v) \leq c(x,y)$, where $c(u,v) = |c(u) - c(v)|$, is a list graph. We show that these graphs are very restricted in nature and enjoy very strong structural properties. Relaxing this condition, we say that a vertex $u$ in a graph $G$ with a labeling $c$ is list-distance consistent if $d(u,v) \leq d(u,w)$ holds for all vertices $v$, $w$ satisfying that $c(u,w) = c(u,v)+1$. The maximum $k$ such that $G$ has a labeling where $k$ vertices are list-distance consistent is the list-distance consistency $\operatorname{ldc}(G)$ of $G$; if $\operatorname{ldc}(G) = |V(G)|$, then $G$ is a local list graph. We prove a structural theorem characterizing local list graphs implying that they are a quite restricted family of graphs; this answers a question of Lennerstad. Furthermore, we investigate the parameter $\operatorname{ldc}(G)$ for various classes of graphs. In particular, we prove that for all $k$, $n$ satisfying $4 \leq k \leq n$ there is a graph $G$ with $n$ vertices and $\operatorname{ldc}(G)=k$, and demonstrate that there are large classes of graphs $G$ satisfying $\operatorname{ldc}(G) = 1$. Indeed, we prove that almost every graph have this property, which implies that graphs $G$ satisfying $\operatorname{ldc}(G) \gt 1$ are in a sense quite rare (let alone local list graphs). We also suggest further variations on the topic of list graphs for future research.
A comprehensive review on the existence of normalized solutions for four classes of nonlinear elliptic equations
(Wydawnictwa AGH, 2025) Chen, Sitong; Tang, Xianhua
This paper provides a comprehensive review of recent results concerning the existence of normalized solutions for four classes of nonlinear elliptic equations: Schrödinger equations, Schrödinger-Poisson equations, Kirchhoff equations, and Choquard equations
Anisotropic singular logistic equations
(Wydawnictwa AGH, 2026) Da Silva, João Pablo Pinheiro; Failla, Giuseppe; Gasiński, Leszek; Papageorgiou, Nikolaos S.
We consider a parametric Dirichlet problem driven by the anisotropic $(p,q)$-Laplacian and a reaction with a singular term and a superdiffusive logistic perturbation. We prove an existence and nonexistence theorem which is global with respect to the parameter $\lambda\gt 0$.We consider a parametric Dirichlet problem driven by the anisotropic $(p,q)$-Laplacian and a reaction with a singular term and a superdiffusive logistic perturbation. We prove an existence and nonexistence theorem which is global with respect to the parameter $\lambda\gt 0$.
Comparison theorems for oscillation and non-oscillation of perturbed Euler type equations
(Wydawnictwa AGH, 2025) Hasil, Petr; Šišoláková, Jiřina; Veselý, Michal
The aim of this paper is to present two comparison theorems. These results enable to describe the oscillation behavior of second order Euler type half-linear differential equations with perturbations in both terms using previously obtained oscillation and non-oscillation criteria. We point out that the comparison theorems are easy to use. This fact is also illustrated by a simple example. In addition, the number of perturbations is arbitrary and the last perturbations can be given by very general continuous functions. Note that the presented results are new even in the case of linear equations.
The automorphism groups of domains and the Greene-Krantz conjecture
(Wydawnictwa AGH, 2025) Krantz, Steven G.
We consider the subject of the automorphism groups of domains in complex space. In particular, we describe and discuss the noted Greene-Krantz conjecture.