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In 1967, Rosa introduced β-labeling which was then popularized by Golomb under the name graceful. Graceful labeling on a graph G is an injective function f:V(G)→{0, 1, 2, ..., |E(G)|} such that, when each edge uv∈E(G) is assigned the label |f(u)-f(v)| the resulting edge labels are distinct. If graph G has graceful labeling then G is called a graceful graph. Rosa also introduced α-labeling on graph G which is a graceful labeling f with an additional condition that there is λ∈{1, 2, ..., |E(G)|} so that for every edge uv∈E(G) where f(u)

Let G = (V (G), E(G)) be a graph with the nonempty vertex set V (G) and the edge set E(G). Let Zn be the group of integers modulo n and let k be a positive integer. A modular irregular labeling of a graph G of order n is an edge k-labeling φ: E(G) → {1, 2,. . ., k}, such that the induced weight function σ: V (G) → Zn defined by σ(v) = Σu∈N(v) φ(uv) (mod n) for every vertex v ∈ V (G) is bijective. The minimum number k such that a graph G has a modular irregular k-labeling is called the modular irregularity strength of a graph G, denoted by ms(G). In this paper, we determine the exact values of the modular irregularity strength of some families of flower graphs, namely rose graphs, daisy graphs and sunflower graphs. [ABSTRACT FROM AUTHOR]

Let G = (V,E) be a simple, connected, and undirected graph, where V and E are the set of vertices and the set of edges of G. An edge magic total labeling on G is a bijection f: V → E → {1, 2, ..., |V | + |E|}, provided that for every uv ∈ E,w(uv) = f(u) + f(v) + f(uv) = K for a constant number K. Such a labeling is said to be a super edge magic total labeling if f(V) = {1, 2,..., |V |} and a b-edge consecutive edge magic total labeling if f(E) = {b + 1, b + 2, ..., b + |E|} with b ≥ 1. In this research, we give sufficient conditions for a graph G having a super edge magic total labeling to have a b-edge consecutive edge magic total labeling. We also give several classes of connected graphs which have both labelings. [ABSTRACT FROM AUTHOR]

Let G(V,E) be a simple graph and f be a bijection f: V ∪ E → {1,2,...,|V| + |E|} where f(V) = {1,2,...,|V |}. For a vertex x ∈ V, define its weight w(x) as the sum of labels of all edges incident with x and the vertex label itself. Then f is called a super vertex local antimagic total (SLAT) labeling if for every two adjacent vertices their weights are different. The super vertex local antimagic total chromatic number χslat (G) is the minimum number of colors taken over all colorings induced by super vertex local antimagic total labelings of G. We classify all trees T that have χslat(T) = 2, present a class of trees that have χslat(T) = 3, and show that for any positive integer n ≥ 2 there is a tree T with χslat(T) = n. [ABSTRACT FROM AUTHOR]

Assume that G(V, E) is a graph with V and E as its vertex and edge sets, respectively. We have G is simple, connected, and undirected. Given a function λ from a union of V and E into a set of k-integers from 1 until k. We call the function λ as a totally irregular total k-labeling if the set of weights of vertices and edges consists of different numbers. For any u ∈ V, we have a weight wt(u) = λ(u) + P uy∈E λ(uy). Also, it is defined a weight wt(e) = λ(u) + λ(uv) + λ(v) for each e = uv ∈ E. A minimum k used in k-total labeling λ is named as a total irregularity strength of G, symbolized by ts(G). We discuss results on ts of some caterpillar graphs in this paper. The results are ts(Sp,2,2,q) = ( p+q−1 2 ) for p, q greater than or equal to 3, while ts(Sp,2,2,2,p) = ( 2p−1/2 ), p ≥ 4. [ABSTRACT FROM AUTHOR]

Let G be a graph with an edge k-coloring γ: E(G) → {1; …; k} (not necessarily proper). A path is called a rainbow path if all of its edges have different colors. The map γis called a rainbow coloring if any two vertices can be connected by a rainbow path. The map γis called a strong rainbow coloring if any two vertices can be connected by a rainbow geodesic. The smallest k for which there is a rainbow k-coloring (resp. strong rainbow k-coloring) on G is called the rainbow connection number (resp. strong rainbow connection number) of G, denoted rc(G) (resp. src(G)). In this paper we generalize the notion of "color codes" that was originally used by Chartrand et al. in their study of the rc and src of complete bipartite graphs, so that it now applies to any connected graph. Using color codes, we prove a new class of lower bounds depending on the existence of sets with common neighbours. Tight examples are discussed, involving the amalgamation of complete graphs, generalized wheel graphs, and a special class of sequential join of graphs. [ABSTRACT FROM AUTHOR]

For a simple graph G, a vertex labeling f : V(G) → {1, 2, ..., k} is called a k-labeling. The weight of a vertex v, denoted by wtf(v) is the sum of all vertex labels of vertices in the closed neighborhood of the vertex v. A vertex k-labeling is defined to be an inclusive distance vertex irregular distance k-labeling of G if for every two different vertices u and v there is wtf(u) ≠ wtf(v). The minimum k for which the graph G has a vertex irregular distance k-labeling is called the inclusive distance vertex irregularity strength of G. In this paper we establish a lower bound of the inclusive distance vertex irregularity strength for any graph and determine the exact value of this parameter for several families of graphs. [ABSTRACT FROM AUTHOR]