Related papers: Distance magic labeling in complete 4-partite grap…

Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a proper total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. We say vertices $u,v\in V$ are sum distinguished if $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. By…

Suppose the vertices of a graph $G$ were labeled arbitrarily by positive integers, and let $Sum(v)$ denote the sum of labels over all neighbors of vertex $v$. A labeling is lucky if the function $Sum$ is a proper coloring of $G$, that is,…

An antimagic labeling of a digraph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to $\{1,2,\cdots,m\}$ such that all $n$ oriented vertex-sums are pairwise distinct, where the oriented vertex-sum of a vertex…

A mapping from the vertex set of a graph G = (V,E) into an interval of integers {0,...,k} is an L(2,1)-labelling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a…

A graph $G$ is said to be $k$-extendable if every matching of size $k$ in $G$ can be extended to a perfect matching of $G$, where $k$ is a positive integer. We say $G$ is $1$-excludable if for every edge $e$ of $G$, there exists a perfect…

For any graph $G=(V,E)$ with maximum degree $\Delta$ and without isolated edges, and a positive integer $r$, by $\chi'_{\Sigma,r}(G)$ we denote the $r$-distant sum distinguishing index of $G$. This is the least integer $k$ for which a…

For a simple graph $G$, the $2$-distance graph, $D_2(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $2$ in the graph $G$. In this paper, we characterize all graphs with connected…

We suggest a new type of problem about distances in graphs and make several conjectures. As a first step towards proving them, we show that for sufficiently large values of n and k, a graph on n vertices that has no three vertices at…

In this paper, we give a characterization of unicyclic graphs with diameter at most 4 which are A-vertex magic. Moreover, let G be a bicyclic graph of diameter 3, then G is group vertex magic if and only if G = M11(0, 0).

We characterize the graphs for which the independence number equals the packing number. As a consequence we obtain simple structural descriptions of the graphs for which (i) the distance-$k$-packing number equals the distance-$2k$-packing…

Given a digraph $D$ with $m$ arcs and a bijection $\tau: A(D)\rightarrow \{1, 2, \ldots, m\}$, we say $(D, \tau)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and no two vertices in $D$ have the same vertex-sum…

The paper presents some bipartite graph $L_{k,n}$, so called $(k,n)$-level graph, that arise by taking $k$-th and $(n-k)$-th levels of $n$-dimensional Boolean algebra. Two results are establised: (1) precise description of a distance (a…

For an $n$-vertex graph $G$, let $z(G;k)$ denote the number of zero forcing sets of size $k$. A conjecture of Boyer et al. asserts that the path $P_n$ maximizes these numbers coefficientwise among all $n$-vertex graphs; equivalently, the…

Given an oriented graph $\overrightarrow{G}$ and $D$ a distance set of $\overrightarrow{G}$, $\overrightarrow{G}$ is $D$-antimagic if there exists a bijective vertex labeling such that the sum of all labels of the $D$-out-neighbors of each…

A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and any list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that…

Tibor Gallai conjectured that the edge set of every connected graph $G$ on $n$ vertices can be partitioned into $\lceil n/2\rceil$ paths. Let $\mathcal{G}_{k}$ be the class of all $2k$-regular graphs of girth at least $2k-2$ that admit a…

Stanley's theorem establishes that for any finite graph $G$, the number $h_G(s)$ of magic labelings with magic sum $s$ can be expressed as a sum of two polynomials in $s$. However, determining the precise form of $h_G(s)$ is generally…

We consider $p$-orientations, which are defined to be orientations of $d$-regular graphs such that every vertex either has in-degree $p$ or out-degree $p$. These generalise the orientations considered in Jaeger's conjecture, where $d=4p+1$.…

Let $m\ge 1$ be an integer and $G$ be a graph with $m$ edges. We say that $G$ has an antimagic orientation if $G$ has an orientation $D$ and a bijection $\tau:A(D)\rightarrow \{1,2,\cdots,m\}$ such that no two vertices in $D$ have the same…

Let $G$ be a connected graph with $|V| = n$ and $|E| = m$. A bijection $f:E\rightarrow \{1,2,...,m\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, $w(u) \neq w(v)$, where $w(u) = \sum_{e \in…