Let G = (V (G),E(G)) be a simple graph and H be a subgraph of G. G admits an H-covering, if every edge in E(G) belongs to at least one subgraph of G that is isomorphic to H. An (a, d)-H-antimagic total labeling of G is a bijection λ: V (G) ∪ E(G) → {1, 2, 3, . . . , |V (G)| + |E(G)|} such that for all subgraphs H′ isomorphic to H, the H′ weights [...] constitute an arithmetic progression a, a+d, a+2d, . . . , a+(n−1)d where a and d are positive integers and n is the number of subgraphs of G isomorphic...

A (p, q)-graph G is (a,d)-edge antimagic total if there exists a bijection f: V(G) ∪ E(G) → {1, 2,...,p + q} such that the edge weights Λ(uv) = f(u) + f(uv) + f(v), uv ∈ E(G) form an arithmetic progression with first term a and common difference d. It is said to be a super (a, d)-edge antimagic total if the vertex labels are {1, 2,..., p} and the edge labels are {p + 1, p + 2,...,p + q}. In this paper, we study the super (a,d)-edge antimagic total labeling of special classes of graphs derived from...

In 1980, Enomoto et al. proposed the conjecture that every tree is a super (a, 0)-edge-antimagic total graph. In this paper, we give a partial sup- port for the correctness of this conjecture by formulating some super (a, d)- edge-antimagic total labelings on a subclass of subdivided stars denoted by T(n, n + 1, 2n + 1, 4n + 2, n5, n6, . . . , nr) for different values of the edge- antimagic labeling parameter d, where n ≥ 3 is odd, nm = 2m−4(4n+1)+1, r ≥ 5 and 5 ≤ m ≤ r.

Enomoto, Llado, Nakamigawa and Ringel (1998) defined the concept of a super (a, 0)-edge-antimagic total labeling and proposed the conjecture that every tree is a super (a, 0)-edge-antimagic total graph. In the support of this conjecture, the present paper deals with different results on super (a, d)-edge-antimagic total labeling of subdivided stars for d ∈ {0, 1, 2, 3}.

A graph $G$ with $p$ vertices and $q$ edges, vertex set $V\left(G\right)$ and edge set $E\left(G\right)$, is said to be super vertex-graceful (in short SVG), if there exists a function pair $(f,f^{+})$ where $f$ is a bijection from $V\left(G\right)$ onto $P$, $f^{+}$ is a bijection from $E\left(G\right)$ onto $Q$, $f^{+}\left((u,v)\right)=f\left(u\right)+f\left(v\right)$ for any $(u,v)\in E\left(G\right)$, $$Q=\left\{\begin{array}{cc}\{\pm 1,\cdots ,\pm \frac{1}{2}q\},\hfill & \text{if}q\text{is}\text{even,}\hfill \\ \{0,\pm 1,\cdots ,\pm \frac{1}{2}(q-1)\},\hfill & \text{if}q\text{is}\text{odd,}\hfill \end{array}\right.$$ and $$P=\left\{\begin{array}{cc}\{\pm 1,\cdots ,\pm \frac{1}{2}p\},\hfill & \text{if}p\text{is}\text{even,}\hfill \\ \{0,\pm 1,\cdots ,\pm \frac{1}{2}(p-1)\},\hfill & \text{if}p\text{is}\text{odd.}\hfill \end{array}\right.$$ We determine here families of unicyclic graphs that are super vertex-graceful.

A graph is called 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. Some constructions of supermagic labellings of regular graphs are described. Supermagic regular complete multipartite graphs and supermagic cubes are characterized.

The notion of a TST-space is introduced and its connection with a parallelogram space is given. The existence of a TST-space is equivalent to the existence of a parallelogram space, which is a new characterization of a parallelogram space. The structure of a TST-space is described in terms of an abelian group.

Let γ(G) and ${\gamma }_{2,2}\left(G\right)$ denote the domination number and (2,2)-domination number of a graph G, respectively. In this paper, for any nontrivial tree T, we show that $\left(2(\gamma \left(T\right)+1)\right)/3\le {\gamma }_{2,2}\left(T\right)\le 2\gamma \left(T\right)$. Moreover, we characterize all the trees achieving the equalities.