arXiv subjects

Marcin Anholcer

Publications and source records attributed to Marcin Anholcer.

23 records · Page 2Linked to original sources

arXiv2012

Distance magic labeling and two products of graphs

Let $G=(V,E)$ be a graph of order $n$. A distance magic labeling of $G$ is a bijection $\ell \colon V\rightarrow {1,...,n}$ for which there exists a positive integer $k$ such that $\sum_{x\in N(v)}\ell (x)=k$ for all $v\in V $, where $N(v)$ is the neighborhood of $v$. We introduce a natural subclass of distance magic graphs. For this class we show that it is closed for the direct product with regular graphs and closed as a second factor for lexicographic product with regular graphs. In addition, we characterize distance magic graphs among direct product of two cycles.

math.CO↗

arXiv2012

Group Irregular Labelings of Disconnected Graphs

We investigate the \textit{group irregularity strength} ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\gr$ of order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums of edge labels at every vertex are distinct. We give the exact values and bounds on $s_g(G)$ for chosen families of disconnected graphs. In addition we present some results for the \textit{modular edge gracefulness} $k(G)$, i.e. the smallest value of $s$ such that there exists a function $f:E(G)\rightarrow \zet_s$ such that the sums of edge labels at every vertex are distinct.

math.CO↗

arXiv2012

Group Irregularity Strength of Connected Graphs

We investigate the group irregularity strength ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\gr$ of order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums of edge labels at every vertex are distinct. We prove that for any connected graph $G$ of order at least 3, $s_g(G)=n$ if $n\neq 4k+2$ and $s_g(G)\leq n+1$ otherwise, except the case of some infinite family of stars.

math.CO↗

arXiv2011

Irregular Labellings of Circulant Graphs

We investigate the \textit{irregularity strength} ($s(G)$) and \textit{total vertex irregularity strength} ($tvs(G)$) of circulant graphs $Ci_n(1,2,...,k)$ and prove that $tvs(Ci_n(1,2,...,k))=\lceil\frac{n+2k}{2k+1}\rceil$, while $s(Ci_n(1,2,...,k))=\lceil\frac{n+2k-1}{2k}\rceil$ except the case when $(n \bmod 4k = 2k+1 \wedge k\bmod 2=1) \vee n=2k+1$ and $s(Ci_n(1,2,...,k))=\lceil\frac{n+2k-1}{2k}\rceil+1$.

math.CO↗

arXiv2011

Total Vertex Irregularity Strength of Forests

We investigate a graph parameter called the total vertex irregularity strength ($tvs(G)$), i.e. the minimal $s$ such that there is a labeling $w: E(G)\cup V(G)\rightarrow \{1,2,..,s\}$ of the edges and vertices of $G$ giving distinct weighted degrees $wt_G(v):=w(v)+\sum_{v\in e \in E(G)}w(e)$ for every pair of vertices of $G$. We prove that $tvs(F)=\lceil (n_1+1)/2 \rceil$ for every forest $F$ with no vertices of degree 2 and no isolated vertices, where $n_1$ is the number of pendant vertices in $F$. Stronger results for trees were recently proved by Nurdin et al.

math.CO↗