A sum graph is a finite simple graph whose vertex set is labeled with distinct positive integers such that two vertices are adjacent if and only if the sum of their labels is itself another label. The spum of a graph $G$ is the minimum difference between the largest and smallest labels in a sum graph consisting of $G$ and the minimum number of additional isolated vertices necessary so that a sum graph labeling exists. We investigate the spum of various families of graphs, namely cycles, paths, and matchings. We introduce the sum-diameter, a modification of the definition of spum that omits the requirement that the number of additional isolated vertices in the sum graph is minimal, which we believe is a more natural quantity to study. We then provide asymptotically tight general bounds on both sides for the sum-diameter, and study its behavior under numerous binary graph operations as well as vertex and edge operations. Finally, we generalize the sum-diameter to hypergraphs.
In "On the super edge graceful trees of even orders," Chung, Lee, Gao, and Schaffer posed the following problem: Characterize trees of diameter 4 which are super edge-graceful. In this paper, we provide super edge-graceful labelings for all caterpillars and even size lobsters of diameter 4 which permit such labelings. We also provide super edge-graceful labelings for several families of odd size lobsters of diameter 4.
For $G$ a simple, connected graph, a vertex labeling $f:V(G)\rightarrow \mathbb{Z}_+$ is called a $\textit{radio labeling of}$ $G$ if it satisfies $|f(u)-f(v)|\geq \operatorname{diam}(G) + 1 - d(u,v)$ for all distinct vertices $u,v\in V(G)$. The $\textit{radio number}$ of $G$ is the minimal span over all radio labelings of $G$. If a bijective radio labeling onto $\{1,2,...,|V(G)|\}$ exists, $G$ is called a $\textit{radio graceful graph}$. We determine the radio number of all diameter $3$ Hamming graphs and show that an infinite subset of them is radio graceful.
Given a graph $G$, a labeling of $G$ is an injective function $f:V(G)\rightarrow\mathbb{Z}_{\ge 0}$. Under the labeling $f$, the label of a vertex $v$ is $f(v)$, and the induced label of an edge $uv$ is $|f(u) - f(v)|$. The labeling $f$ is graceful if the labels of the vertices are $\{0, 1, \ldots , |V(G)| - 1\}$, and the induced labels of the edges are distinct. The graph $G$ is graceful if it has a graceful labeling. The Graceful Tree Conjecture, introduced by Kotzig in the late 1960's, states that all trees are graceful. It is an open problem whether every diameter-6 tree has a graceful lab…
A radio labeling of a connected graph $G$ is a function $c:V(G) \to \mathbb Z_+$ such that for every two distinct vertices $u$ and $v$ of $G$ $$\text{distance}(u,v)+|c(u)-c(v)|\geq 1+ \text{diameter}(G).$$ The radio number of a graph $G$ is the smallest integer $M$ for which there exists a labeling $c$ with $c(v)\leq M$ for all $v\in V(G)$. The radio number of graphs of order $n$ and diameter $n-1$, i.e., paths, was determined in \cite{paths}. Here we determine the radio numbers of all graphs of order $n$ and diameter $n-2$.
For $k\in\mathbb{Z}^+$ and $G$ a simple connected graph, a $k$-radio labeling $f:V_G\to\Z^+$ of $G$ requires all pairs of distinct vertices $u$ and $v$ to satisfy $|f(u)-f(v)|\geq k+1-d(u,v)$. When $k=1$, this requirement gives rise to the familiar labeling known as vertex coloring for which each vertex of a graph is labeled so that adjacent vertices have different "colors". We consider $k$-radio labelings of $G$ when $k=\diam(G)$. In this setting, no two vertices can have the same label, so graphs that have radio labelings of consecutive integers are one extreme on the spectrum of possibiliti…
We answer two questions of Shamik Ghosh in the negative. We show that there exists a lobster tree of diameter less than 6 which accepts no alpha-labeling with two central vertices labeled by the critical number and the maximum vertex label. We also show a simple example of a tree of diameter 4, with an even degree central vertex which does not accept a maximum label in any graceful labeling.
For any positive integer $h$, a graph $G=(V,E)$ is said to be $h$-magic if there exists a labeling $l:E(G)\to \mathbb{Z}_h -\{0\} $ such that the induced vertex set labeling $\ l^+ : V(G) \to \mathbb{Z}_h \ $ defined by $$ l^+ (v)=\sum_{uv \in E(G)} \ l(uv) $$ is a constant map. The integer-magic spectrum of a graph $G$, denoted by $IM(G)$, is the set of all $h \in \mathbb{N} $ for which $G$ is $h$-magic. So far, only the integer-magic spectra of trees of diameter at most five have been determined. In this paper, we determine the integer-magic spectra of trees of diameter six and higher.
A tree $T$ on $2^n$ vertices is called set-sequential if the elements in $V(T)\cup E(T)$ can be labeled with distinct nonzero $(n+1)$-dimensional $01$-vectors such that the vector labeling each edge is the component-wise sum modulo $2$ of the labels of the endpoints. It has been conjectured that all trees on $2^n$ vertices with only odd degree are set-sequential (the "Odd Tree Conjecture"), and in this paper, we present progress toward that conjecture. We show that certain kinds of caterpillars (with restrictions on the degrees of the vertices, but no restrictions on the diameter) are set-sequ…
Let $G$ be a simple connected graph. For any two vertices $u$ and $v$, let $d(u,v)$ denote the distance between $u$ and $v$ in $G$. A radio-$k$-labeling of $G$ for a fixed positive integer $k$ is a function $f$ which assigns to each vertex a non-negative integer label such that for every two vertices $u$ and $ v $ in $G$, $|f(u)-f(v)| \geq k - d(u,v) +1$. The span of $f$ is the difference between the largest and smallest labels of $f(V)$. The radio-$k$-number of a graph $G$, denoted by $rn_k(G)$, is the smallest span among all radio-$k$-labelings admitted by $G$. A cycle $C_n$ has diameter $d=…
For a connected graph $G$, let $diam(G)$ and $d(u,v)$ denote the diameter of $G$ and distance between $u$ and $v$ in $G$. A radio labeling of a graph $G$ is a mapping $φ: V(G) \rightarrow \{0,1,2,...\}$ such that $|φ(u)-φ(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u, v$ of $G$. The span of $φ$ is defined as span($φ$) = $\max\{|φ(u)-φ(v)| : u, v \in V(G)\}$. The radio number $rn(G)$ of $G$ is defined as $rn(G)$ = $\min\{$span($φ$) : $φ$ is a radio labeling of $G\}$. In this paper, we determine the radio number for middle graph of paths.
Let $G$ be a finite, connected, undirected graph with diameter $diam(G)$ and $d(u,v)$ denote the distance between $u$ and $v$ in $G$. A radio labeling of a graph $G$ is a mapping $f: V(G) \rightarrow \{0,1,2,...\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u, v$ of $G$. The radio number of $G$, denoted by $rn(G)$, is the smallest integer $k$ such that $G$ has a radio labeling $f$ with $\max\{f(v) : v \in V(G)\} = k$. In this paper, we determine the radio number for three families of trees obtained by taking graph operation on a given tree or a famil…
Let $\mathbb{N}$ be the set of positive integers. A radio labeling of a graph $G$ is a mapping $φ: V(G) \rightarrow \mathbb{N} \cup \{0\}$ such that the inequality $|φ(u)-φ(v)| \geq diam(G) + 1 - d(u,v)$ holds for every pair of distinct vertices $u,v$ of $G$, where $diam(G)$ and $d(u,v)$ are the diameter of $G$ and distance between $u$ and $v$ in $G$, respectively. The radio number $rn(G)$ of $G$ is the smallest number $k$ such that $G$ has radio labeling $φ$ with $\max\{φ(v) : v \in V(G)\}$ = $k$. Das et al. [Discrete Math. $\mathbf{340}$(2017) 855-861] gave a technique to find a lower bound …
A radio labeling of a graph $G$ is a mapping $\vp : V(G) \rightarrow \{0, 1, 2,...\}$ such that $|\vp(u)-\vp(v)|\geq \diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$, where $\diam(G)$ and $d(u,v)$ are the diameter of $G$ and distance between $u$ and $v$ in $G$, respectively. The radio number $\rn(G)$ of $G$ is the smallest number $k$ such that $G$ has radio labeling with $\max\{\vp(v):v \in V(G)\}$ = $k$. In this paper, we slightly improve the lower bound for the radio number of graphs given by Das \emph{et al.} in [5] and, give necessary and sufficient condition to achi…
