The metric dimension problem was first introduced in 1975 by Slater [12], and independently by Ha... more The metric dimension problem was first introduced in 1975 by Slater [12], and independently by Harary and Melter [6] in 1976; however the problem for hypercube was studied (and solved asymptotically) much earlier in 1963 by Erdős and Rényi [4]. A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G.
In this paper, we define D-magic labelings for oriented graphs where D is a distance set. In part... more In this paper, we define D-magic labelings for oriented graphs where D is a distance set. In particular, we label the vertices of the graph with distinct integers {1,2,…,|V(G)|} in such a way that the sum of all the vertex labels that are a distance in D away from a given vertex is the same across all vertices. We give some results related to the magic constant, construct a few infinite families of D-magic graphs, and examine trees, cycles, and multipartite graphs. This definition grew out of the definition of D-magic (undirected) graphs. This paper explores some of the symmetries we see between the undirected and directed version of D-magic labelings.
For a set of distances $D$, a graph $G$ of order $n$ is said to be $D-$magic if there exists a bi... more For a set of distances $D$, a graph $G$ of order $n$ is said to be $D-$magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots, n\}$ and a constant $k$ such that for any vertex $x$, $\sum_{y\in N_D(x)} f(y) =k$, where $N_D(x)=\{y|d(y,x)=j, j\in D\}$. In this paper we shall find sets of distances $D$s, such that the hypercube is $D-$magic. We shall utilise well-known properties of (bipartite) distance-regular graphs to construct the $D-$magic labelings.
A graph $G$ is said to be distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldot... more A graph $G$ is said to be distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , v\}$ and a constant {\sf k} such that for any vertex $x$, $\sum_{y\in N(x)} f(y) ={\sf k}$, where $N_(x)$ is the set of all neighbours of $x$. In this paper we shall study distance magic labelings of graphs obtained from four graph products: cartesian, strong, lexicographic, and cronecker. We shall utilise magic rectangle sets and magic column rectangles to construct the labelings.
For a set of distances D, a graph G on n vertices is said to be D-magic if there exists a bijecti... more For a set of distances D, a graph G on n vertices is said to be D-magic if there exists a bijection f : V → {1, 2, . . . , n} and a constant k such that for any vertex x, ∑ y∈ND(x) f(y) = k, where ND(x) = {y|d(x, y) = i, i ∈ D} is the Dneighbourhood set of x. In this paper we utilize spectra of graphs to characterize strongly regular graphs which are D-magic, for all possible distance sets D. In addition, we provide necessary conditions for distance regular graphs of diameter 3 to be {1}-magic.
A graph G admits an H-covering if every edge of G belongs to a subgraph isomorphic to a given gra... more A graph G admits an H-covering if every edge of G belongs to a subgraph isomorphic to a given graph H. G is said to be H-magic if there exists a bijection f:V(G)∪E(G)→{1,2,…,|V(G)|+|E(G)|} such that wf(H′)=∑v∈V(H′)f(v)+∑e∈E(H′)f(e) is a constant, for every subgraph H′ isomorphic to H. In particular, G is said to be H-supermagic if f(V(G))={1,2,…,|V(G)|}. When H is isomorphic to a complete graph K2, an H-(super)magic labeling is an edge-(super)magic labeling. Suppose that G admits an F-covering and H-covering for two given graphs F and H. We define G to be (F,H)-sim-(super)magic if there exists a bijection f′ that is simultaneously F-(super)magic and H-(super)magic. In this paper, we consider (K2,H)-sim-(super)magic where H is isomorphic to three classes of graphs with varied symmetry: a cycle which is symmetric (both vertex-transitive and edge-transitive), a star which is edge-transitive but not vertex-transitive, and a path which is neither vertex-transitive nor edge-transitive. We...
The study of locating-chromatic number of a graph initiated by Chartrand et al. [5] is only limit... more The study of locating-chromatic number of a graph initiated by Chartrand et al. [5] is only limited for connected graphs. In 2014, Welyyanti et al. extended this notion so that the locating-chromatic number can also be applied to disconnected graphs. Let c be a k-coloring of a disconnected graph H(V, E) and ∏ = {C1,C2, …, Ck} be the partition of V (H) induced by c, where Ci is the set of all vertices receiving color i. The color code c∏(v) of a vertex v ∈ H is the ordered k-tuple (d(v,C1), d(v,C2), …, d(v,Ck)), where d(v,Ci) = min{d(v, x)|x ∈ Ci} and d(v,Ci) < ∞ for all i ∈ [1, k]. If all vertices of H have distinct color codes, then c is called a locating-coloring of H. The locating-chromatic number of H, denoted by χ′L(H), is the smallest k such that H admits a locating-coloring with k colors, otherwise we say that χ′L(H)=∞. In this paper, we determine locating-chromatic number of a graph with two components where each component has the locating-chromatic number 3.
Let G(V, E) be a simple graph. For a labeling ϕ : V (G) ∪ E(G) → {1, 2, …, k} the weight of a ver... more Let G(V, E) be a simple graph. For a labeling ϕ : V (G) ∪ E(G) → {1, 2, …, k} the weight of a vertex x is defined as wt(x) = ϕ(x) + ∑y∈N(x) ϕ (xy), where N(x) is the set of neighbors of x and y. The labeling ϕ is called a vertex irregular total k-labeling if for every pair of distinct vertices x and y we have wt(x) ≠ wt(y). The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G and is denoted by tvs(G). In this paper, we determine total vertex irregularity strengths of trees with maximum degree four and a subdivision of a double-star.
The metric dimension problem was first introduced in 1975 by Slater [12], and independently by Ha... more The metric dimension problem was first introduced in 1975 by Slater [12], and independently by Harary and Melter [6] in 1976; however the problem for hypercube was studied (and solved asymptotically) much earlier in 1963 by Erdős and Rényi [4]. A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G.
In this paper, we define D-magic labelings for oriented graphs where D is a distance set. In part... more In this paper, we define D-magic labelings for oriented graphs where D is a distance set. In particular, we label the vertices of the graph with distinct integers {1,2,…,|V(G)|} in such a way that the sum of all the vertex labels that are a distance in D away from a given vertex is the same across all vertices. We give some results related to the magic constant, construct a few infinite families of D-magic graphs, and examine trees, cycles, and multipartite graphs. This definition grew out of the definition of D-magic (undirected) graphs. This paper explores some of the symmetries we see between the undirected and directed version of D-magic labelings.
For a set of distances $D$, a graph $G$ of order $n$ is said to be $D-$magic if there exists a bi... more For a set of distances $D$, a graph $G$ of order $n$ is said to be $D-$magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots, n\}$ and a constant $k$ such that for any vertex $x$, $\sum_{y\in N_D(x)} f(y) =k$, where $N_D(x)=\{y|d(y,x)=j, j\in D\}$. In this paper we shall find sets of distances $D$s, such that the hypercube is $D-$magic. We shall utilise well-known properties of (bipartite) distance-regular graphs to construct the $D-$magic labelings.
A graph $G$ is said to be distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldot... more A graph $G$ is said to be distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , v\}$ and a constant {\sf k} such that for any vertex $x$, $\sum_{y\in N(x)} f(y) ={\sf k}$, where $N_(x)$ is the set of all neighbours of $x$. In this paper we shall study distance magic labelings of graphs obtained from four graph products: cartesian, strong, lexicographic, and cronecker. We shall utilise magic rectangle sets and magic column rectangles to construct the labelings.
For a set of distances D, a graph G on n vertices is said to be D-magic if there exists a bijecti... more For a set of distances D, a graph G on n vertices is said to be D-magic if there exists a bijection f : V → {1, 2, . . . , n} and a constant k such that for any vertex x, ∑ y∈ND(x) f(y) = k, where ND(x) = {y|d(x, y) = i, i ∈ D} is the Dneighbourhood set of x. In this paper we utilize spectra of graphs to characterize strongly regular graphs which are D-magic, for all possible distance sets D. In addition, we provide necessary conditions for distance regular graphs of diameter 3 to be {1}-magic.
A graph G admits an H-covering if every edge of G belongs to a subgraph isomorphic to a given gra... more A graph G admits an H-covering if every edge of G belongs to a subgraph isomorphic to a given graph H. G is said to be H-magic if there exists a bijection f:V(G)∪E(G)→{1,2,…,|V(G)|+|E(G)|} such that wf(H′)=∑v∈V(H′)f(v)+∑e∈E(H′)f(e) is a constant, for every subgraph H′ isomorphic to H. In particular, G is said to be H-supermagic if f(V(G))={1,2,…,|V(G)|}. When H is isomorphic to a complete graph K2, an H-(super)magic labeling is an edge-(super)magic labeling. Suppose that G admits an F-covering and H-covering for two given graphs F and H. We define G to be (F,H)-sim-(super)magic if there exists a bijection f′ that is simultaneously F-(super)magic and H-(super)magic. In this paper, we consider (K2,H)-sim-(super)magic where H is isomorphic to three classes of graphs with varied symmetry: a cycle which is symmetric (both vertex-transitive and edge-transitive), a star which is edge-transitive but not vertex-transitive, and a path which is neither vertex-transitive nor edge-transitive. We...
The study of locating-chromatic number of a graph initiated by Chartrand et al. [5] is only limit... more The study of locating-chromatic number of a graph initiated by Chartrand et al. [5] is only limited for connected graphs. In 2014, Welyyanti et al. extended this notion so that the locating-chromatic number can also be applied to disconnected graphs. Let c be a k-coloring of a disconnected graph H(V, E) and ∏ = {C1,C2, …, Ck} be the partition of V (H) induced by c, where Ci is the set of all vertices receiving color i. The color code c∏(v) of a vertex v ∈ H is the ordered k-tuple (d(v,C1), d(v,C2), …, d(v,Ck)), where d(v,Ci) = min{d(v, x)|x ∈ Ci} and d(v,Ci) < ∞ for all i ∈ [1, k]. If all vertices of H have distinct color codes, then c is called a locating-coloring of H. The locating-chromatic number of H, denoted by χ′L(H), is the smallest k such that H admits a locating-coloring with k colors, otherwise we say that χ′L(H)=∞. In this paper, we determine locating-chromatic number of a graph with two components where each component has the locating-chromatic number 3.
Let G(V, E) be a simple graph. For a labeling ϕ : V (G) ∪ E(G) → {1, 2, …, k} the weight of a ver... more Let G(V, E) be a simple graph. For a labeling ϕ : V (G) ∪ E(G) → {1, 2, …, k} the weight of a vertex x is defined as wt(x) = ϕ(x) + ∑y∈N(x) ϕ (xy), where N(x) is the set of neighbors of x and y. The labeling ϕ is called a vertex irregular total k-labeling if for every pair of distinct vertices x and y we have wt(x) ≠ wt(y). The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G and is denoted by tvs(G). In this paper, we determine total vertex irregularity strengths of trees with maximum degree four and a subdivision of a double-star.
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