Note on group distance magic graphs G[C4]G[C_4]

A \emph{group distance magic labeling} or a \gr-distance magic labeling of a graph $G(V,E)$ with $|V | = n$ is an injection $f$ from $V$ to an Abelian group \gr of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of every vertex $x \in V$ is equal to the same element \mu \in \gr, called the magic constant. In this paper we will show that if $G$ is a graph of order $n=2^{p}(2k+1)$ for some natural numbers $p$, $k$ such that \deg(v)\equiv c \imod {2^{p+1}} for some constant $c$ for any $v\in V(G)$, then there exists an \gr-distance magic labeling for any Abelian group \gr for the graph $G[C_4]$. Moreover we prove that if \gr is an arbitrary Abelian group of order $4n$ such that \gr \cong \zet_2 \times\zet_2 \times \gA for some Abelian group \gA of order $n$, then exists a \gr-distance magic labeling for any graph $G[C_4]$

Oberwolfach rectangular table negotiation problem

AbstractWe completely solve certain case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Let H(k,3) be a bipartite graph with bipartition X={x1,x2,…,xk},Y={y1,y2,…,yk} and edges x1y1,x1y2,xkyk−1,xkyk, and xiyi−1,xiyi,xiyi+1 for i=2,3,…,k−1. We completely characterize all complete bipartite graphs Kn,n that can be factorized into factors isomorphic to G=mH(k,3), where k is odd and mH(k,3) is the graph consisting of m disjoint copies of H(k,3)

Incarcerated Adults’ Perceptions Of Remaining Opiate Free Upon Release

Abstract Incarcerated adults face the challenge of remaining opiate free after release from incarceration. Despite services offered in jail substance use disorder disproportionately affects approximately half of incarcerated adults. There has been little research into the perceptions related to incarcerated adults experiences to remain opiate free upon release from jail. This descriptive phenomenological project helped elucidate this matter by understanding the lived experiences of five adults who had undergone opiate use treatment while incarcerated, were released, and re-used opiates. The nurse investigator conducted face-to-face interviews with these participants. Seven open-ended questions related to their perceptions and multiple probing questions resulted in rich, deep, and robust data which were analyzed by constant comparative analysis and coded into themes and subthemes. The revealed themes included: Anticipation of Staying Opiate Free, Difficulties in Staying Opiate Free, Benefits to Being Opiate Free, and Needed to Remain Opiate Free. The findings revealed that these incarcerated adults experience a plethora of perceived challenges to remaining opiate free upon release. An increased understanding of motivations, behaviors, and perspectives can better inform nursing practice and target strategies aimed at bringing about behavioral change by mitigating the myriad of vulnerability factors and perceived challenges to remain opiate free after release

Regular handicap tournaments of high degree

A handicap distance antimagic labeling of a graph $G=(V,E)$ with $n$ vertices is a bijection ${f}: V\to \{ 1,2,\ldots ,n\}$ with the property that ${f}(x_i)=i$ and the sequence of the weights $w(x_1),w(x_2),\ldots,w(x_n)$ (where $w(x_i)=\sum\limits_{x_j\in N(x_i)}f(x_j)$) forms an increasing arithmetic progression with difference one. A graph $G$ is a {\em handicap distance antimagic graph} if it allows a handicap distance antimagic labeling. We construct $(n-7)$-regular handicap distance antimagic graphs for every order $n\equiv2\pmod4$ with a few small exceptions. This result complements results by Kov\'a\v{r}, Kov\'a\v{r}ov\'a, and Krajc~[P. Kov\'a\v{r}, T. Kov\'a\v{r}ov\'a, B. Krajc, On handicap labeling of regular graphs, manuscript, personal communication, 2016] who found such graphs with regularities smaller than $n-7$

A note on incomplete regular tournaments with handicap two of order n≡8(mod 16)

A $d$-handicap distance antimagic labeling of a graph $G=(V,E)$ with $n$ vertices is a bijection $f:V\to \{1,2,\ldots ,n\}$ with the property that $f(x_i)=i$ and the sequence of weights $w(x_1),w(x_2),\ldots,w(x_n)$ (where $w(x_i)=\sum_{x_i x_j\in E}f(x_j)$) forms an increasing arithmetic progression with common difference $d$. A graph $G$ is a $d$-handicap distance antimagic graph if it allows a $d$-handicap distance antimagic labeling. We construct a class of $k$-regular $2$-handicap distance antimagic graphs for every order $n\equiv8\pmod{16}$, $n\geq56$ and $6\leq k\leq n-50$

Distance Magic Graphs - a Survey

Let <i>G = (V;E)</i> be a graph of order n. A bijection <i>f : V → {1, 2,...,n} </i>is called <i>a distance magic labeling </i>of G if there exists a positive integer k such that <i>Σ f(u) = k </i> for all <i>v ε V</i>, where <i>N(v)</i> is the open neighborhood of v. The constant k is called the magic constant of the labeling f. Any graph which admits <i>a distance magic labeling </i>is called a distance magic graph. In this paper we present a survey of existing results on distance magic graphs along with our recent results,open problems and conjectures.DOI : http://dx.doi.org/10.22342/jims.0.0.15.11-2