In this paper, we consider Skolem (vertex) labellings and present (hooked) Skolem labellings for generalised Dutch windmills whenever such labellings exist. Specifically, we show that generalised Dutch windmills with more than two cycles cannot be Skolem labelled and that those composed of two cycles of lengths m and n, n ≥ m, cannot be Skolem labelled if and only if n−m ≡ 3, 5 (mod 8) and m is...

A simple undirected graph H is called a sum graph if there is a labeling L of the vertices of H into distinct positive integers such that any two vertices u and v of H are adjacent if and only if there is a vertex w with label L(w) = L(u) + L(v). The sum number (G) of a graph G = (V; E) is the least integer r such that the graph H consisting of G and r isolated vertices is a sum graph. It is cl...

A binary vertex coloring (labeling) f : V (G) → Z2 of a graph G is said to be friendly if the number of vertices labeled 0 is almost the same as the number of vertices labeled 1. This friendly labeling induces an edge labeling f∗ : E(G) → Z2 defined by f∗(uv) = f(u)f(v) for all uv ∈ E(G). Let ef (i) = |{uv ∈ E(G) : f∗(uv) = i}| be the number of edges of G that are labeled i. Product-cordial ind...

A k-circular-distance-two labeling (or k-c-labeling) of a simple graph G is a vertex-labeling, using the labels 0, 1, 2, · · · , k − 1, such that the “circular difference” (mod k) of the labels for adjacent vertices is at least two, and for vertices of distance-two apart is at least one. The σ-number, σ(G), of a graph G is the minimum k of a k-c-labeling of G. For any given positive integers n ...

Let f : V (G) → {1, 2,..., |V (G)|} be a bijection, and let us denote S = f(u) + f(v) D |f(u) − f(v)| for every edge uv in E(G). f' the induced labeling, by vertex labeling f, defined as E(G) {0, 1} such that any E(G), (uv)=1 if gcd(S, D)=1, (uv)=0 otherwise. ef' (0) (1) number of edges labeled with 0 1 respectively. is SD-prime cordial |ef' (1)| ≤ G graph it admits labeling. In this paper, we ...

We consider undirected simple finite graphs. The sets of vertices and edges of a graph G are denoted by V (G) and E(G), respectively. For a graph G, we denote by δ(G) and η(G) the least degree of a vertex of G and the number of connected components of G, respectively. For a graph G and an arbitrary subset V0 ⊆ V (G) G[V0] denotes the subgraph of the graph G induced by the subset V0 of its verti...

For a graph G = (V,E) and a binary labeling f : V (G) → Z2, let vf (i) = |f−1(i)|. The labling f is said to be friendly if |vf (1)−vf (0)| ≤ 1. Any vertex labeling f : V (G) → Z2 induces an edge labeling f∗ : E(G) → Z2 defined by f∗(xy) = |f(x)− f(y)|. Let ef (i) = |f∗−1(i)|. The friendly index set of the graph G, denoted by FI(G), is defined by FI(G) = {|ef (1)− ef (0)| : f is a friendly verte...

The sets of vertices and edges of an undirected, simple, finite, connected graph G are denoted by V (G) and E(G), respectively. An arbitrary nonempty finite subset of consecutive integers is called an interval. An injective mapping φ : E(G) → {1, 2, . . . , |E(G)|} is called a labeling of the graph G. If G is a graph, x is its arbitrary vertex, and φ is its arbitrary labeling, then the set SG(x...

A vertex (edge) irregular total k-labeling ? of a graph G is labeling the vertices and edges with labels from set {1,2,...,k} in such way that any two different (edges) have distinct weights. Here, weight x sum label all incident x, whereas an edge to edge. The minimum k for which has called irregularity strength G. In this paper, we are dealing infinite classes convex polytopes generated by pr...

A 1-vertex-magic vertex labeling of a graph G(V,E) with p vertices is a bijection f from the vertex set V (G) to the integers 1, 2, . . . , p with the property that there is a constant k such that at any vertex x the sum ∑ f(x) taken over all neighbors of x is k. In this paper, we study the 1-vertex-magic vertex labelings of two families of disconnected graphs, namely a disjoint union of m copi...