On H-irregular reflexive labeling of graph

For any non-trivial abelian group A under addition a graph $G$ is said to be $A$-textit{magic} if there exists a labeling $f:E(G) rightarrow A-{0}$ such that, the vertex labeling $f^+$ defined as $f^+(v) = sum f(uv)$ taken over all edges $uv$ incident at $v$ is a constant. An $A$-textit{magic} graph $G$ is said to be $Z_k$-magic graph if the group $A$ is $Z_k$ the group of integers modulo $k$. These $Z_k$-magic graphs are referred to as $k$-textit{magic} graphs. In this paper we prove that the total graph, flower graph, generalized prism graph, closed helm graph, lotus inside a circle graph, $Godotoverline{K_m}$, $m$-splitting graph of a path and $m$-shadow graph of a path are $Z_k$-magic graphs.

Let G be a (p, q) graph. Let k be an integer with 2 ≤ k ≤ p and f from V (G) to the set {1, 2, . . . , k} be a map. For each edge uv, assign the label |f(u) − f(v)|. The function f is called a k-difference cordial labeling of G if |νf (i) − vf (j)| ≤ 1 and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labelled with x (x ∈ {1, 2 . . . , k}), ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate the 3-difference cordial labeling of wheel, helms, flower graph, sunflower graph, lotus inside a circle, closed helm, and double wheel.

L ( 2 , 1 ) -labeling is a graph coloring model inspired by a frequency assignment in telecommunication. It asks for such a labeling of vertices with nonnegative integers that adjacent vertices get labels that differ by at least 2 and vertices in distance 2 get different labels. It is known that for any k ⩾ 4 it is NP-complete to determine if a graph has a L ( 2 , 1 ) -labeling with no label greater than k. In this paper we present a new bound on complexity of an algorithm for finding an optimal L ( 2 , 1 ) -labeling (i.e. an L ( 2 , 1 ) -labeling in which the largest label is the least possible). We improve the upper complexity bound of the algorithm from O ⁎ ( 3.5616 n ) to O ⁎ ( 3.2361 n ) . Moreover, we establish a lower complexity bound of the presented algorithm, which is Ω ⁎ ( 3.0739 n ) .

Research in graph theory has lead to one of the important area called labeling of graphs. There are different types of labeling such as graceful labeling, magic labeling, edge-graceful labeling, prime labeling, radio labeling, harmonious labeling etc., In this paper, we investigate the existence of edge graceful labeling for the Corona graph Cn⊙K2¯ (n is odd n ≥ 3). Also we have defined a flower graph which does not admit an edge-graceful labeling.

An $$L(j,k)$$L(j,k)-labeling of a graph is a vertex labeling such that the difference of the labels of any two adjacent vertices is at least $$j$$j and that of any two vertices of distance $$2$$2 is at least $$k$$k. The minimum span of all $$L(j,k)$$L(j,k)-labelings of $$G$$G is denoted by $$\lambda _k^j(G)$$?kj(G). Lin and Lam (Discret Math 308:3805---3815, 2008) provided an upper bound of $$\lambda _1^2(K_m \times C_n)$$?12(Km×Cn) when $$K_m \times C_n$$Km×Cn is the direct product of a complete graph $$K_m$$Km and a cycle $$C_n$$Cn. And they found the exact value of $$\lambda _1^2(K_m \times C_n)$$?12(Km×Cn) for some $$m$$m and $$n$$n. In this paper, we obtain an upper bound and a lower bound of $$\lambda _1^1(K_m \times C_n)$$?11(Km×Cn). As a consequence we compute $$\lambda _1^1(K_m \times C_n)$$?11(Km×Cn) when $$n$$n is even or $$n\ge 4m+1$$n?4m+1.

An $L(j,k)$-$labeling$ of a graph is a vertex labeling such that the difference between the labels of adjacent vertices is at least $j$ and that between vertices separated by a distance 2 is at least $k$. The minimum of the spans of all $L(j,k)$-labelings of $G$ is denoted by $\lambda_k^j(G)$. Recently, Haque and Jha [16] proved that if $G$ is a multiple direct product of complete graphs, then $\lambda_k^j(G)$ coincides with the trivial lower bound $(N-1)k$, where $N$ is the order of $G$ and $\frac{j}{k}$ is within a certain bound. In this paper, we suggest a new labeling method for such a graph $G$. With this method, we extend the range of $\frac{j}{k}$ such that $\lambda_k^j(G)= (N-1)k$ holds. Moreover, we obtain the upper bound of $\lambda_k^j(G)$ for the remaining cases in the range $\frac{j}{k}$.

L(2,1)-colorings and Irreducible No-hole Colorings of Cartesian Product of Graphs

[Formula: see text]-labeling problem ([Formula: see text]-[Formula: see text]) is an important topic in discrete mathematics due to its various applications, like in frequency assignment in mobile communication systems, signal processing, circuit design, etc. [Formula: see text]-[Formula: see text] is a special case of [Formula: see text]-[Formula: see text]. An [Formula: see text]-labeling ([Formula: see text]) of a graph [Formula: see text] is a mapping [Formula: see text] such that [Formula: see text] if and only if [Formula: see text], [Formula: see text] if [Formula: see text] or 3, where [Formula: see text] is the distance between the nodes [Formula: see text] and [Formula: see text]. In this work, we have determined the upper bound of [Formula: see text] for interval graph (IG) and obtained a tighter upper bound which is [Formula: see text]. Also, we proposed an efficient algorithm to label any IG by [Formula: see text] and also computed the time complexity of the proposed algorithm.

For any simple undirected graph G(V, E), a map f : V ⋃ E → {1, 2, …, k} such that for any different edges xy and x’y’ their weights are distinct is called an edge irregular total k-labeling. The weight of edge xy is defined as the sum of edge label of xy, vertex label of x and vertex label of y. The minimum k for which the graph G has an edge irregular total k-labeling is called the total edge irregularity strength of G and is denoted by tes(G). In this paper, we determine the exact value of the total edge irregularity strength of odd arithmetic book graph Bn(C3, 5, 7, …2n+1) and even arithmetic book graph Bn(C4, 6, 8, …, 2n+2) of n sheets. We found that the tes of odd arithmetic book graph Bn(C3, 5, 7, …, 2n+1) of n sheets is equal to the ceiling function of and the tes of even arithmetic book graph Bn(C4, 6, 8, …, 2n+2) is equal to the ceiling function of .

Let G = (V (G), E(G)) be a graph with q edges. A function f is called harmonious labeling of graph G if f: V →¨{0, 1, 2,…, q-1} is injective and the induced function f*: E → {0, 1, 2,…, q} defined as f*(uv) = (f(u) + f(v))(mod q) is bijective. A graph which admits harmonious labeling is called harmonious graph. In this paper we prove that the jewel graph, triangular ladder graph, special flower graph, duplicating all the vertex of mK1, in P2+mK1, T(Pn)Kcm are harmonious graphs.

We introduce the notion of total distance irregular labeling, called the local total distance irregular labeling. All edges and vertices are labeled with positive integers 1 to k such that the weight calculated at the vertices induces a vertex coloring if two adjacent vertices has different weight. The weight of a vertex $u\in V(G)$ is defined as the sum of the labels of all vertices adjacent and edges incident to $u$ (distance $1$ from $u$). The minimum cardinality of the largest label over all such irregular assignment is called the local total distance irregularity strength, denoted by $tdis_l(G)$. In this paper, we established the lower bound of the local total distance irregularity strength of graphs $G$ and determined exact values of some classes of graphs namely path, cycle, star, bipartite complete, fan and sun graph.

A graph G of order p and size q is called super edge-magic if there exists a bijective function f from V(G) U E(G) to {1, 2, 3, ..., p+q} such that f(x) + f(xy) + f(y) is a constant for every edge $xy \in E(G)$ and f(V(G)) = {1, 2, 3, ..., p}. The super edge-magic deficiency of a graph G is either the smallest nonnegative integer n such that G U nK_1 is super edge-magic or +~ if there exists no such integer n. In this paper, we study the super edge-magic deficiency of join product graphs. We found a lower bound of the super edge-magic deficiency of join product of any connected graph with isolated vertices and a better upper bound of the super edge-magic deficiency of join product of super edge-magic graphs with isolated vertices. Also, we provide constructions of some maximal graphs, ie. super edge-magic graphs with maximal number of edges.