A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that the (p, 1)-total labelling number of every 1-planar graph G is at most Δ(G) + 2p − 2 provided that Δ(G) ≥ 8p+4 or Δ(G) ≥ 6p+2 and g(G) ≥ 4. As a consequence, the well-known (p, 1)-total labelling conjecture has been confirmed for some 1-planar graphs.

A graph $G$ of order $n$ is said to be a prime graph if its vertices can be labeled with the first $n$ positive integers in such a way that the labels of any two adjacent vertices in $G$ are relatively prime. If such a labeling on $G$ exists then it is called a prime labeling. In this paper we seek prime labeling for union of tadpole graphs. We derive a necessary condition for the existence of prime labelings of graphs that are union of tadpole graphs and further show that the condition is also sufficient...

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.

We study the structure of path-like trees. In order to do this, we introduce a set of trees that we call expandable trees. In this paper we also generalize the concept of path-like trees and we call such generalization generalized path-like trees. As in the case of path-like trees, generalized path-like trees, have very nice labeling properties.

Let G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑uv∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give...

Martin Bača et al. [2] introduced the problem of determining the total vertex irregularity strengths of graphs. In this paper we discuss how the addition of new edge affect the total vertex irregularity strength.

A graph G is called (H;k)-vertex stable if G contains a subgraph isomorphic to H ever after removing any of its k vertices. Q(H;k) denotes the minimum size among the sizes of all (H;k)-vertex stable graphs. In this paper we complete the characterization of $(K_{m,n};1)$-vertex stable graphs with minimum size. Namely, we prove that for m ≥ 2 and n ≥ m+2, $Q(K_{m,n};1)=mn+m+n$ and $K_{m,n}*K₁$ as well as $K_{m+1,n+1}-e$ are the only $(K_{m,n};1)$-vertex stable graphs with minimum size, confirming the conjecture of Dudek and Zwonek.

Let $D$ be an oriented graph of order $n$ and size $m$. A $\gamma$-labeling of $D$ is a one-to-one function $fV\left(D\right)\to \{0,1,2,...,m\}$ that induces a labeling $f^{\text{'}}E\left(D\right)\to \{\pm 1,\pm 2,...,\pm m\}$ of the arcs of $D$ defined by $f^{\text{'}}\left(e\right)=f\left(v\right)-f\left(u\right)$ for each arc $e=(u,v)$ of $D$. The value of a $\gamma$-labeling $f$ is $\mathrm{v}al\left(f\right)={\sum }_{e\in E\left(G\right)}{f}^{\text{'}}\left(e\right).$ A $\gamma$-labeling of $D$ is balanced if the value of $f$ is 0. An oriented graph $D$ is balanced if $D$ has a balanced labeling. A graph $G$ is orientably balanced if $G$ has a balanced orientation. It is shown that a connected graph $G$ of order $n\ge 2$ is orientably balanced unless $G$ is a tree, $n\equiv 2\left(mod4\right)$, and every vertex of...

Let G be a graph of order n and size m. A γ-labeling of G is a one-to-one function f:V(G) → 0,1,2,...,m that induces a labeling f’: E(G) → 1,2,...,m of the edges of G defined by f’(e) = |f(u)-f(v)| for each edge e = uv of G. The value of a γ-labeling f is $val\left(f\right)={\Sigma }_{e\in E\left(G\right)}{f}^{\text{'}}K\left(e\right)$. The maximum value of a γ-labeling of G is defined as $va{l}_{max}\left(G\right)=maxval\left(f\right):fisa\gamma -labelingofG$; while the minimum value of a γ-labeling of G is $va{l}_{min}\left(G\right)=minval\left(f\right):fisa\gamma -labelingofG$; The values $va{l}_{max}\left(S_{p,q}\right)$ and $va{l}_{min}\left(S_{p,q}\right)$ are determined for double stars $S_{p,q}$. We present characterizations of connected graphs G of order n for which $va{l}_{min}\left(G\right)=n$ or $va{l}_{min}\left(G\right)=n+1$.

An edge ranking of a graph is a labeling of edges using positive integers such that all paths connecting two edges with the same label visit an intermediate edge with a higher label. An edge ranking of a graph is optimal if the number of labels used is minimum among all edge rankings. As the problem of finding optimal edge rankings for general graphs is NP-hard [12], it is interesting to concentrate on special classes of graphs and find optimal edge rankings for them efficiently. Apart from trees...