Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2016
Volume
Vol. 36
Number
No. 5
Description
Journal Volume
Opuscula Mathematica
Vol. 36 (2016)
Projects
Pages
Articles
Criticality indices of 2-rainbow domination of paths and cycles
(2016) Bouchou, Ahmed; Blidia, Mostafa
A $2$-rainbow dominating function of a graph $G\left(V(G),E(G)\right)$ is a function $f$ that assigns to each vertex a set of colors chosen from the set ${1,2}$ so that for each vertex with $f(v)=\emptyset$ we have ${\textstyle\bigcup_{u\in N(v)}} f(u)=\{1,2\}$. The weight of a 2RDF $f$ is defined as $w\left( f\right)={\textstyle\sum\nolimits_{v\in V(G)}} |f(v)|$. The minimum weight of a $2$RDF is called the $2$-rainbow domination number of $G$, denoted by $\gamma_{2r}(G)$. The vertex criticality index of a $2$-rainbow domination of a graph $G$ is defined as $ci_{2r}^{v}(G)=(\sum\nolimits_{v\in V(G)}(\gamma_{2r}\left(G\right) -\gamma_{2r}\left( G-v\right)))/\left\vert V(G)\right\vert$, the edge removal criticality index of a $2$-rainbow domination of a graph $G$ is defined as $ci_{2r}^{-e}(G)=(\sum\nolimits_{e\in E(G)}(\gamma_{2r}\left(G\right)-\gamma_{2r}\left( G-e\right)))/\left\vert E(G)\right\vert$ and the edge addition of a $2$-rainbow domination criticality index of $G$ is defined as $ci_{2r}^{+e}(G)=(\sum\nolimits_{e\in E(\overline{G})}(\gamma_{2r}\left(G\right)-\gamma_{2r}\left( G+e\right)))/\left\vert E(\overline{G})\right\vert$, where $\overline{G}$ is the complement graph of $G$. In this paper, we determine the criticality indices of paths and cycles.
Edge subdivision and edge multisubdivision versus some domination related parameters in generalized corona graphs
(2016) Dettlaff, Magda; Raczek, Joanna Patrycja; Yero, Ismael González
Given a graph $G=(V,E)$, the subdivision of an edge $e=uv\in E(G)$ means the substitution of the edge $e$ by a vertex x and the new edges $ux$ and $xv$. The domination subdivision number of a graph $G$ is the minimum number of edges of $G$ which must be subdivided (where each edge can be subdivided at most once) in order to increase the domination number. Also, the domination multisubdivision number of $G$ is the minimum number of subdivisions which must be done in one edge such that the domination number increases. Moreover, the concepts of paired domination and independent domination subdivision (respectively multisubdivision) numbers are defined similarly. In this paper we study the domination, paired domination and independent domination (subdivision and multisubdivision) numbers of the generalized corona graphs.
On one oscillatory criterion for the second order linear ordinary differential equations
(2016) Grigorian, Gevorg Avagovich
The Riccati equation method is used to establish an oscillatory criterion for second order linear ordinary differential equations. An oscillatory condition is obtained for the generalized Hill's equation. By means of examples the obtained result is compared with some known oscillatory criteria.
M₂-edge colorings of dense graphs
(2016) Ivančo, Jaroslav
An edge coloring $\varphi$ of a graph $G$ is called an $\mathrm{M}_i$-edge coloring if $|\varphi(v)|\leq i$ for every vertex $v$ of $G$, where $\varphi(v)$ is the set of colors of edges incident with $v$. Let $\mathcal{K}_i(G)$ denote the maximum number of colors used in an $\mathrm{M}_i$-edge coloring of $G$. In this paper we establish some bounds of $\mathcal{K}_2(G)$, present some graphs achieving the bounds and determine exact values of $\mathcal{K}_2(G)$ for dense graphs.
Existence and boundary behavior of positive solutions for a Sturm-Liouville problem
(2016) Masmoudi, Syrine; Zermani, Samia
In this paper, we discuss existence, uniqueness and boundary behavior of a positive solution to the following nonlinear Sturm-Liouville problem $\begin{aligned}&\frac{1}{A}(Au^{\prime })^{\prime }+a(t)u^{\sigma}=0\;\;\text{in}\;(0,1),\\ &\lim\limits_{t\to 0}Au^{\prime}(t)=0,\quad u(1)=0,\end{aligned}$ where $\sigma \lt 1$, $A$ is a positive differentiable function on $(0,1)$ and $a$ is a positive measurable function in $(0,1)$ satisfying some appropriate assumptions related to the Karamata class. Our main result is obtained by means of fixed point methods combined with Karamata regular variation theory.