Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2018
Volume
Vol. 38
Number
No. 6
Description
Journal Volume
Opuscula Mathematica
Vol. 38 (2018)
Projects
Pages
Articles
Improved bounds for solutions of φ-Laplacians
(Wydawnictwa AGH, 2018) Arriagada, Waldo; Huentutripay, Jorge
In this short paper we prove a parametric version of the Harnack inequality for $\phi$-Laplacian equations. In this sense, the estimates are optimal and represent an improvement of previous bounds for this kind of operators.
On signed arc total domination in digraphs
(Wydawnictwa AGH, 2018) Asgharsharghi, Leila; Khodkar, Abdollah; Sheikholeslami, S. M.
Let $D=(V,A)$ be a finite simple digraph and $N(uv)=\{u^{\prime}v^{\prime}\neq uv \mid u=u^{\prime}\text{ or }v=v^{\prime}\}$ be the open neighbourhood of uv in $D$. A function $f: A\rightarrow \{-1, +1\}$ is said to be a signed arc total dominating function (SATDF) of $D$ if $\sum _{e^{\prime}\in N(uv)}f(e^{\prime})\geq 1$ holds for every arc $uv\in A$. The signed arc total domination number $\gamma^{\prime}_{st}(D)$ is defined as $\gamma^{\prime}_{st}(D)= \operatorname{min}\{\sum_{e\in A}f(e)\mid f \text{ is an SATDF of }D\}$. In this paper we initiate the study of the signed arc total domination in digraphs and present some lower bounds for this parameter.
On locally irregular decompositions of subcubic graphs
(Wydawnictwa AGH, 2018) Baudon, Olivier; Bensmail, Julien; Hocquard, Hervé; Senhaji, Mohammed; Sopena, Éric
A graph $G$ is locally irregular if every two adjacent vertices of $G$ have different degrees. A locally irregular decomposition of $G$ is a partition $E_1,\dots,E_k$ of $E(G)$ such that each $G[E_{i}]$ is locally irregular. Not all graphs admit locally irregular decompositions, but for those who are decomposable, in that sense, it was conjectured by Baudon, Bensmail, Przybyło and Woźniak that they decompose into at most 3 locally irregular graphs. Towards that conjecture, it was recently proved by Bensmail, Merker and Thomassen that every decomposable graph decomposes into at most 328 locally irregular graphs. We here focus on locally irregular decompositions of subcubic graphs, which form an important family of graphs in this context, as all non-decomposable graphs are subcubic. As a main result, we prove that decomposable subcubic graphs decompose into at most 5 locally irregular graphs, and only at most 4 when the maximum average degree is less than $\frac{12}{5}$. We then consider weaker decompositions, where subgraphs can also include regular connected components, and prove the relaxations of the conjecture above for subcubic graphs.
Zig-zag facial total-coloring of plane graphs
(Wydawnictwa AGH, 2018) Czap, Július; Jendroľ, Stanislav; Voigt, Margit
In this paper we introduce the concept of zig-zag facial total-coloring of plane graphs. We obtain lower and upper bounds for the minimum number of colors which is necessary for such a coloring. Moreover, we give several sharpness examples and formulate some open problems.
Essential norm of an integral-type operator from ω-Bloch spaces to μ-Zygmund spaces on the unit ball
(Wydawnictwa AGH, 2018) Du, Juntao; Zhu, Xiangling
In this paper, we give an estimate for the essential norm of an integral-type operator from $\omega$-Bloch spaces to $\mu$-Zygmund spaces on the unit ball.