For a graph G = (V, E), a function f:V(G) → 1,2, ...,k is a k-ranking if f(u) = f(v) implies that every u - v path contains a vertex w such that f(w) > f(u). A k-ranking is minimal if decreasing any label violates the definition of ranking. The arank number, ${\psi }_r\left(G\right)$, of G is the maximum value of k such that G has a minimal k-ranking. We completely determine the arank number of the Cartesian product Kₙ ☐ Kₙ, and we investigate the arank number of Kₙ ☐ Kₘ where n > m.

Chartrand et al. (2004) have given an upper bound for the nearly antipodal chromatic number $a{c}^{\text{'}}\left(P_n\right)$ as $\left(\genfrac{}{}{0pt}{}{n-2}{2}\right)+2$ for $n\ge 9$ and have found the exact value of $a{c}^{\text{'}}\left(P_n\right)$ for $n=5,6,7,8$. Here we determine the exact values of $a{c}^{\text{'}}\left(P_n\right)$ for $n\ge 8$. They are $2p^2-6p+8$ for $n=2p$ and $2p^2-4p+6$ for $n=2p+1$. The exact value of the radio antipodal number $ac\left(P_n\right)$ for the path $P_n$ of order $n$ has been determined by Khennoufa and Togni in 2005 as $2p^2-2p+3$ for $n=2p+1$ and $2p^2-4p+5$ for $n=2p$. Although the value of $ac\left(P_n\right)$ determined there is correct, we found a mistake in the proof of the lower bound when $n=2p$ (Theorem $6$). However,...

A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ℓ from V to an Abelian group Γ of order n such that the weight $$w\left(x\right)={\sum }_{y\in N_G\left(x\right)}\ell \left(y\right)$$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ℤp-distance magic. Moreover...

For a connected graph G of order n ≥ 3, let f: E(G) → ℤₙ be an edge labeling of G. The vertex labeling f’: V(G) → ℤₙ induced by f is defined as $f^{\text{'}}\left(u\right)={\sum }_{v\in N\left(u\right)}f\left(uv\right)$, where the sum is computed in ℤₙ. If f’ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A modular edge-graceful labeling f of G is nowhere-zero if f(e) ≠ 0 for all e ∈ E(G) and in this case, G is a nowhere-zero modular edge-graceful graph. It is shown that a connected graph G of order n ≥ 3 is nowhere-zero...

A graph $G$ is called ${(k,d)}^{*}$-choosable if for every list assignment $L$ satisfying $\left|L\right(v\left)\right|=k$ for all $v\in V\left(G\right)$, there is an $L$-coloring of $G$ such that each vertex of $G$ has at most $d$ neighbors colored with the same color as itself. In this paper, it is proved that every toroidal graph without chordal 7-cycles and adjacent 4-cycles is ${(4,1)}^{*}$-choosable.

The additive hereditary property of linear forests is characterized by the existence of average labellings.

Is it possible to label the edges of Kₙ with distinct integer weights so that every Hamilton cycle has the same total weight? We give a local condition characterizing the labellings that witness this question's perhaps surprising affirmative answer. More generally, we address the question that arises when "Hamilton cycle" is replaced by "k-factor" for nonnegative integers k. Such edge-labellings are in correspondence with certain vertex-labellings, and the link allows us to determine (up to a constant...

A proper coloring $c:V\left(G\right)\to \{1,2,...,k\}$, $k\ge 2$ of a graph $G$ is called a graceful $k$-coloring if the induced edge coloring $c^{\text{'}}:E\left(G\right)\to \{1,2,...,k-1\}$ defined by $c^{\text{'}}\left(uv\right)=|c\left(u\right)-c\left(v\right)|$ for each edge $uv$ of $G$ is also proper. The minimum integer $k$ for which $G$ has a graceful $k$-coloring is the graceful chromatic number ${\chi }_g\left(G\right)$. It is known that if $T$ is a tree with maximum degree $\Delta$, then ${\chi }_g\left(T\right)\le \lceil \frac{5}{3}\Delta \rceil$ and this bound is best possible. It is shown for each integer $\Delta \ge 2$ that there is an infinite class of trees $T$ with maximum degree $\Delta$ such that ${\chi }_g\left(T\right)=\lceil \frac{5}{3}\Delta \rceil$. In particular, we investigate for each integer $\Delta \ge 2$ a...

In this paper, we improve the result by Harper on the lower bound of the bandwidth of connected graphs. In addition, we prove that considerating the interior boundary and the exterior boundary when estimating the bandwidth of connected graphs gives the same results.

As introduced by F. Harary in 1994, a graph $G$ is said to be an $integral$$sum$$graph$ if its vertices can be given a labeling $f$ with distinct integers so that for any two distinct vertices $u$ and $v$ of $G$, $uv$ is an edge of $G$ if and only if $f\left(u\right)+f\left(v\right)=f\left(w\right)$ for some vertex...

S.M. Lee proposed the conjecture: for any n > 1 and any permutation f in S(n), the permutation graph P(Pₙ,f) is graceful. For any integer n > 1 and permutation f in S(n), we discuss the gracefulness of the permutation graph P(Pₙ,f) if $f={\prod }_{k=0}^{l-1}(m+2k,m+2k+1)$, and ${\prod }_{k=0}^{l-1}(m+4k,m+4k+2)(m+4k+1,m+4k+3)$ for any positive integers m and l.

A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (consecutive) positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. We characterize magic line graphs of general graphs and describe some class of supermagic line graphs of bipartite graphs.

A graph is called magic (supermagic) if it admits a labeling of the edges by pairwise different (and consecutive) integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper we characterize magic joins of graphs and we establish some conditions for magic joins of graphs to be supermagic.