Group A Cordial Labeling

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said to be a $k$-super graceful labeling of $G$. We say $G$ is $k$-super graceful if it admits a $k$-super graceful labeling. In this paper, we study the $k$-super gracefulness of some regular and bi-regular graphs.

Labeling of graphs is the procedure of assigning numbers to the nodes, lines, or both in accordance with an applicable rule. In this study, we demonstrate that the complete graph K_n (n=3) is a compact mean-labeled graph. We also explored graph〖 K 〗_4is compact mean-labeled graphs

Let G be a (p,q) graph and A be a group. Let f : V (G) → A be a function. The order of u ∈ A is the least positive integer n such that u n = e. We denote the order of u by o(u). For each edge uv assign the label 1 if (o(u), o(v)) = 1 or 0 otherwise. f is called a group A Cordial labeling if |vf (a) − vf (b)| ≤ 1 and |ef (0) − ef (1)| ≤ 1, where vf (x) and ef (n) respectively denote the number of vertices labeled with an element x and number of edges labeled with n(n = 0, 1). A graph which admits a group A Cordial labeling is called a group A Cordial graph. In this paper we define group {1,−1, i,−i} Cordial graphs and prove that Hypercube Qn = Qn−1 × K2, Book Bn = Sn × K2, n-sided prism Prn = Cn × K2 and Pn × K3 are all group {1,−1, i,−i} Cordial for all n.

Let G be a (p,q)graph and A be a group. Let f : V (G) → A be a function. The order of a ∈ A is the least positive integer n such that a N = e. We denote the order of a by o(a). For each edge uv assign the label 1 if (o(f (u)), o(f (v))) = 1or 0 otherwise.

The present authors are motivated by two research articles "Divisor Cordial Graphs" by Varatharajan et al. and "Square Divisor Cordial Graphs" by Murugesan et al. We introduce the concept of cube divisor cordial labeling. A cube divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2, ,| |} V … such that an edge e uv = is assigned the label 1 if 3

A cube divisor cordial labeling of a graph G with vertex set V (G) is a bijection f from V (G) to {1, 2,. .. , |V (G)|} such that an edge e = uv is assigned the label 1 if [f (u)] 3 |f (v) or [f (v)] 3 |f (u) and the label 0 otherwise, then |e f (0) − e f (1)| ≤ 1. A graph which admits a cube divisor cordial labeling is called a cube divisor cordial graph. In this paper we discuss cube divisor cordial labeling of some standard graphs such as path, cycle, wheel, flower and fan.

→A which satisfies the conditions |v f (a)-v f (b)|≤1 and |e f (a)-e f (b)|≤1, for all a,b v A, when the edge e=uv is labeled as f(u)*f(v). Where v f (a) is the number of vertices with label a and e f (a) is the number of edges with label a. If we consider an Abelian Group < A,* >=< Z k , + k > then it is called k-cordial labeling. In this research paper we proved that Z-P n , braid graph B(n), triangular ladder TL n and irregular quadrilateral snake I(QS n) are 4cordial for all n.

We discuss here k-cordial labeling of prism for all odd k. We prove that prisms P m ×C k , P m ×C k+1 , P m ×C k+3 are k-cordial for all odd k and m ≥ 2. In addition to this we prove that all the Prisms P m ×C 2k-1 are k-cordial for all odd k, m ≥ 2 and m ≠ tk; t ≥ 1.

A graph that admits a ∧ cordial labeling is called a ∧ cordial graph (CCG). In this paper, we proved that cycle C&lt;sub&gt;n&lt;/sub&gt; (n is even), bistar B&lt;sub&gt;m,n&lt;/sub&gt;, P&lt;sub&gt;m&lt;/sub&gt; ⊙ P&lt;sub&gt;n&lt;/sub&gt; and Helm are ∧ cordial graphs.

Let G = (V,E) be a graph with p vertices and q edges. A Cup (V) cordial labeling of a Graph G with vertex set V is a bijection from V to {0,1} such that if each edge uv is assigned the label with the condition the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1 and the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. The graph that admits a V cordial labeling is called a V cordial graph (CCG). In this paper, we proved that Cycle C n (n : even), Bistar B m , n , n m P P Θ and Helm are V cordial graphs.

and the label 0 otherwise, then |e f (0) -e f (1)| ≤ 1. A graph which admits a cube divisor cordial labeling is called a cube divisor cordial graph. In this paper we discuss cube divisor cordial labeling of some standard graphs such as path, cycle, wheel, flower and fan.

Let G be a (p,q) graph and A be a group. Let f : V (G) → A be a function. The order of u ∈ A is the least positive integer n such that u n = e. We denote the order of u by o(u). For each edge uv assign the label 1 if (o(u), o(v)) = 1 or 0 otherwise. f is called a group A Cordial labeling if |vf (a) − vf (b)| ≤ 1 and |ef (0) − ef (1)| ≤ 1, where vf (x) and ef (n) respectively denote the number of vertices labeled with an element x and number of edges labeled with n(n = 0, 1). A graph which admits a group A Cordial labeling is called a group A Cordial graph. In this paper we define group {1,−1, i,−i} Cordial graphs and prove that Hypercube Qn = Qn−1 × K2, Book Bn = Sn × K2, n-sided prism Prn = Cn × K2 and Pn × K3 are all group {1,−1, i,−i} Cordial for all n.

Let G=(V(G),E(G)) be a simple, finite and undirected graph of order p and size q. For k&gt; 1, a bijection f: V(G)∪ E(G) →{k, k+1, k+2, ..., k+p+q-1} such that f(uv)= |f(u) - f(v)| for every edge uv∈ E(G) is said to be a k-super graceful labeling of G. We say G is k-super graceful if it admits a k-super graceful labeling. In this paper, we study the k-super gracefulness of some regular and bi-regular graphs.

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said to be a $k$-super graceful labeling of $G$. We say $G$ is $k$-super graceful if it admits a $k$-super graceful labeling. In this paper, we study the $k$-super gracefulness of some regular and bi-regular graphs.

An integer cordial labeling of a graph G(V, E) is an injective map f from V to or  as p is even or odd, which induces an edge labeling f * : E → {0, 1} defined by f * (uv) = 1 if f(u) + f(v) ≥ 0 is positive and 0 otherwise such that the number of edges labeled with1and the number of edges labeled with 0 differ atmost by 1. If a graph has integer cordial labeling, then it is called integer cordial graph . In this paper, we introduce the concept of integer cordial labeling and prove that some standard graphs are integer cordial.

The total product cordial labeling is a variant of cordial labeling. We introduce an edge analogue product cordial labeling as a variant of total product cordial labeling and name it as total edge product cordial labeling. Unlike to total product cordial labeling the roles of vertices and edges are interchanged in total edge product cordial labeling. We investigate several results on this newly defined concept.

In this paper, the face edge product cordial labeling of planar graphs Tn for even n, M(Pn) for odd n, the star of cycle Cn for odd n, the graph G obtained by joining two copies of planar graph G by a path of arbitrary length and the path union of k copies of cycle Cn except for odd k and even n are presented. We also discussed the total face edge product cordial labeling of fn, Wn and the star of cycle Cn and face product cordial labeling of the graph G obtained by joining two copies of planar graph G by a path of arbitrary length and the path union of k copies of cycle Cn except for odd k and even n. AMS subject classifications : 05C78

A sum divisor cordial labeling of a graph G with vertex set V is a bijection f from V(G) to {1,2,…, |V(G)|} such that an edge uv is assigned the label 1 if 2 divides f(u)+f(v) and 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph with a sum divisor cordial labeling is called a sum divisor cordial graph. In this paper, we investigate the sum divisor cordial labeling of switching of a pendent vertex in path P n , switching of any vertex in cycle C

In this paper we introduce two new labeling types, namely face product cordial labeling and total face product cordial labeling and also investigate the face product cordial labeling of fan, M(P n), S′(P n) except for odd n, T(P n), T n , H n , S n except for even n and one vertex union of mC n and C mn and total face product cordial labeling of H n , S n and W n .

A mapping f : V (G) → {0, 1, 2} is called 3-product cordial labeling
if |vf (i) − vf (j)| ≤ 1 and |ef (i) − ef (j)| ≤ 1 for any i, j ∈ {0, 1, 2},
where vf (i) denotes the number of vertices labeled with i, ef (i) denotes
the number of edges xy with f(x)f(y) ≡ i(mod 3). A graph with 3-
product cordial labeing is called 3-product cordial graph. In this paper
we establish that switching of an apex vertex in closed helm, double
fan, book graph K1,n × K2 and permutation graph P(K2 + mK1, I)
are 3-product cordial graphs.

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , p} be a function. For each edge uv, assign the label |f (u)− f (v)|. f is called a difference cordial labeling if f is a one to one map and |ef (0)− ef (1)| ≤ 1 where ef (1) and ef (0) denote the number of edges labeled with 1 and not labeled with 1 respectively. A graph which admits a difference cordial labeling is called a difference cordial graph. In this paper we investigate the difference cordial labeling behaviour of K2 +mK1, K n + 2K2, Sunflower grpah, Lotus inside a circle, Pyramid, Permutation graphs.

Let G be a (p,q) graph and A be a group. Let f : V (G) → A be a function. The order of a ∈ A is the least positive integer n such that a n = e. We denote the order of a by o(a). For each edge uv assign the label 1 if (o(f(u)), o(f(v))) = 1or 0 otherwise. f is called a group A Cordial labeling if |v f (a)−v f (b)| ≤ 1 and |e f (0)−e f (1)| ≤ 1, where v f (x) and e f (n) respectively denote the number of vertices labeled with an element x and number of edges labeled with n(n = 0, 1). A graph which admits a group A Cordial labeling is called a group A Cordial graph. The Splitting graph of G, S′(G) is obtained from G by adding for each vertex v of G, a new vertex v′ so that v′ is adjacent to every vertex that is adjacent to v. Note that if G is a (p, q) graph then S′(G) is a (2p, 3q) graph. In this paper we prove that Splitting graphs of Star S ' (K 1,n), Fan S′(F n), Comb S′(P n ΘK 1), Ladder S′(L n), Friendship graph S′(C n (3)), Umbrella graph S′(U n,n) and Book S′(B n) are group {1,−1, i,−i} Cordial for every n.

Let G be a (p,q) graph and A be a group. Let f : V (G) → A be a function. The order of u ∈ A is the least positive integer n such that un = e. We denote the order of u by o(u). For each edge uv assign the label 1 if (o(u), o(v)) = 1 or 0 otherwise. f is called a group A Cordial labeling if |vf (a) − vf (b)| ≤ 1 and |ef (0) − ef (1)| ≤ 1, where vf (x) and ef (n) respectively denote the number of vertices labeled with an element x and number of edges labeled with n(n = 0, 1). A graph which admits a group A Cordial labeling is called a group A Cordial graph. In this paper we define group {1,−1, i,−i} Cordial graphs and prove that Hypercube Qn = Qn−1 × K2, Book Bn = Sn × K2, n-sided prism Prn = Cn × K2 and Pn × K3 are all group {1,−1, i,−i} Cordial for all n.

Let (,) be a finite simple graph. Let be a function from the edge set to { , }. For each vertex ∈ , define () = ∑{ (): ∈ }(). The function is called an − labeling of G, if the number of edges labeled 0 and the number of edges labeled 1 differs by at most one and the number of vertices labeled 0 and the number of vertices labeled 1 differs by at most one. A graph that admits − labeling is said to E-Cordial. In this paper, we prove that Odd Snakes and () , that is one vertex union of copies of , for t-even and-odd are E-Cordial.

A mapping f : V (G) → {0, 1, 2} is called 3-product cordial labeling if |v f (i) − v f (j)| ≤ 1 and |e f (i) − e f (j)| ≤ 1 for any i, j ∈ {0, 1, 2}, where v f (i) denotes the number of vertices labeled with i, e f (i) denotes the number of edges xy with f (x)f (y) ≡ i(mod 3). A graph with 3product cordial labeing is called 3-product cordial graph. In this paper we establish that switching of an apex vertex in closed helm, double fan, book graph K 1,n × K 2 and permutation graph P (K 2 + mK 1 , I) are 3-product cordial graphs.

A mapping f : V (G) → {0, 1, 2} is called a 3-product cordial labeling if |v f (i) − v f (j)| ≤ 1 and |e f (i) − e f (j)| ≤ 1 for any i, j ∈ {0, 1, 2}, where v f (i) denotes the number of vertices labeled with i, e f (i) denotes the number of edges xy with f (x)f (y) ≡ i (mod 3). A graph with a 3-product cordial labeling is called a 3-product cordial graph. In this paper, we establish that the duplicating arbitrary vertex in cycle Cn, duplicating arbitrarily edge in cycle Cn, duplicating arbitrary vertex in wheel Wn, Ladder Ln, Triangular Ladder T Ln and the graph W (1) n : W (2) n : • • • : W (k) n are 3-product cordial.

Let G be a connected graph and d(x, y) be the distance between the vertices x and y. A subset of vertices W = {w1,. .. , w k } is called a resolving set for G if for every two distinct vertices x, y ∈ V (G), there is a vertex wi ∈ W such that d(x, wi) = d(y, wi). A resolving set containing a minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension dim(G). A family of connected graphs G is said to be a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in G. In this paper, we show that generalized Petersen graphs P (n, 2), antiprisms A n and Harary graphs H 4,n for n ≡ 1(mod 4) are families of regular graphs with constant metric dimension and raise some questions in a more general setting.

We characterize strongly edge regular product graphs and find the edgebalanced index sets of complete bipartite graphs without a perfect matching, the direct product K n ×K 2. We also prove a lemma that is helpful to determine the edge-balanced index sets of regular graphs.

We introduce the concept of the total edge fixing edge-to-vertex detour setof a connected graph . Let e be an edge of a graph . A set is called an edge fixing edge-to-vertex detour set of a connected graph  if every edge of lies on an detour, where . The edge fixing edge-to-vertex detour number  of  is the minimum cardinality of its edge fixing edge-to-vertex detour sets and any edge fixing edge-to-vertex detour set of cardinality    is an -set of Connected graphs of order p with edge fixing edge-to-vertex detour number 1 or  are characterized. Theedge fixing edge-to-vertex detour number for some standard graphs are determined.  It is shown that for every pair of positive integers with 2  there exists a connected graph  such that  and  for some edge .

Let G be a (p,q)graph and A be a group. Let f : V (G) → A be a function. The order of a ∈ A is the least positive integer n such that a N = e. We denote the order of a by o(a). For each edge uv assign the label 1 if (o(f (u)), o(f (v))) = 1or 0 otherwise. f is called a group A Cordial labeling if |v F (a) − v F (b)| ≤ 1 and |e F (0) − e F (1)| ≤ 1, where v F (x) and e F (n) respectively denote the number of vertices labeled with an element x and number of edges labeled with n(n = 0, 1). A graph which admits a group A Cordial labeling is called a group A Cordial graph.

Let G be a (p,q) graph and A be a group. Let f : V (G) → A be a function. The order of u ∈ A is the least positive integer n such that un = e. We denote the order of u by o(u). For each edge uv assign the label 1 if (o(u), o(v)) = 1 or 0 otherwise. f is called a group A Cordial labeling if |vf (a) − vf (b)| ≤ 1 and |ef (0) − ef (1)| ≤ 1, where vf (x) and ef (n) respectively denote the number of vertices labeled with an element x and number of edges labeled with n(n = 0, 1). A graph which admits a group A Cordial labeling is called a group A Cordial graph. In this paper we define group {1,−1, i,−i} Cordial graphs and prove that Hypercube Qn = Qn−1 × K2, Book Bn = Sn × K2, n-sided prism Prn = Cn × K2 and Pn × K3 are all group {1,−1, i,−i} Cordial for all n.

A sum divisor cordial labeling of a graph G with vertex set V is a bijection f from V(G) to {1,2,…, |V(G)|} such that an edge uv is assigned the label 1 if 2 divides f(u)+f(v) and 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph with a sum divisor cordial labeling is called a sum divisor cordial graph. In this paper, we investigate the sum divisor cordial labeling of switching of a pendent vertex in path P n , switching of any vertex in cycle C

In this paper we introduce two new labeling types, namely face product cordial labeling and total face product cordial labeling and also investigate the face product cordial labeling of fan, M(P n), S′(P n) except for odd n, T(P n), T n , H n , S n except for even n and one vertex union of mC n and C mn and total face product cordial labeling of H n , S n and W n .

In this paper, the face edge product cordial labeling of planar graphs T n for even n, M(P n) for odd n, the star of cycle C n for odd n, the graph G obtained by joining two copies of planar graph G by a path of arbitrary length and the path union of k copies of cycle C n except for odd k and even n are presented. We also discussed the total face edge product cordial labeling of f n , W n and the star of cycle C n and face product cordial labeling of the graph G obtained by joining two copies of planar graph G by a path of arbitrary length and the path union of k copies of cycle C n except for odd k and even n. AMS subject classifications : 05C78 Keywords-face edge product cordial graph, total face edge product cordial graph, face product cordial graph. I. INTRODUCTION By a graph, we mean a simple, finite, planar and undirected unless otherwise specified. A (p,q) planar graph G means a graph G = (V,E), where V is the set of vertices with |V| = p, E is the set of edges with |E| = q and F is the set of interior faces of G with |F| = number of interior faces of G, for terms not defined here, we refer to Harary [4]. For standard terminology and notations related to graph labeling, we refer to Gallian [3]. In [2], Cahit introduce the concept of cordial labeling of graph. The concept of product cordial labeling of a graph was introduced by Sundaram et.al., [9]. In [10], Sundaram et al. also have introduced total product cordial labeling of graph. The concept of signed product cordial labeling was introduced by Baskar Babujee et al. [1]. In [11], Vaidya et al. introduced the concept of edge product cordial labeling of graph. The edge product cordial labeling of various types of graph are presented in [12]. The concept of total edge product cordial labeling is introduced by Vaidya et al. [13]. Sedlacek [8] defined a graph to be magic if it had an edge-labeling, with range the real numbers, such that the sum of the labels around any vertex equals some constant, independent of the choice of vertex. In 1983, Lih [7] introduced magic labelings of planar graphs where labels extended to faces as well as edges and vertices, an idea which he traced back to 13th century Chinese roots. Motivated by the concept of various types of product cordial labeling and magic labeling, we introduce face product cordial labeling, total face product cordial labeling, face edge product cordial labeling, total face edge product cordial labeling, face signed product cordial labeling and total face signed product cordial labeling of graph. In [5], Lawrence et al. proved the face signed product cordial labeling of the Pl n , n 5 except n 0 (mod 4) and the graph Pl m,n , m,n 3. Here we also prove the total face signed product cordial labeling of the Pl n , n 4 and the graph Pl m,n , m, n 3. In [6], Lawrence et al. proved the face product cordial labeling of fan, M(P n), S(P n) except for odd n, T(P n), T n , H n , S n except for even n and one vertex union of mC n and C mn. Here we also proved the total face product cordial labeling of H n , S n and W n. As every edge product cordial graph does not admit face edge product cordial labeling it is very interesting to find out graphs or graph families which admit face edge product cordial labeling. The brief summaries of definition which are necessary for the present investigation are provided below.

In this paper we define total magic cordial (TMC) and total sequential cordial (TSC) labellings which are weaker versions of magic and simply sequential labellings of graphs. Based on these definitions we have given several results on TMC and TSC graphs.