Edge Xy Research Articles

Odd graceful labeling on the Ilalang graph (Sn , 3) it’s variation

A graph called odd graceful graph if there is an injective mapping f:V(G) → {0,1,2,…,2q − 1} such that for each edge xy 6 E(G) labeled by |f(u) − f(v)|, and the results willform a label on the edge {1,3,5, …,2q − 1} which is bijective. Variabel q is the number of edges. This paper purpose is to construct the formula of odd graceful labelling on the ilalang graph (Sn , 3) and it’s variation.

An application of super mean and magic graphs labeling on cryptography system

An edge-magic total (EMT) labeling on a simple graph G with p vertices and q edges is a bijection f: V∪E→{1, 2, …, p + q} such that for each edge xy of G, f(x) + f(xy) + f(y) = k, for a fixed positive integer k. Moreover, an EMT labeling f is a super edge-magic total labeling (SEMT) if f(V) = {1, 2, …, p}. Furthermore, a graph G is called a super mean (SM) graph if there exists an injective function f from the vertices of G to {0,1,2,…,q} such that when each edge uv is labeled with (f(u) +f(v))/2 when f(u) +f(v) is even, and (f(u) +f(v) + 1)/2 when f(u) +f(v) is odd, then the resulting edge labels are distinct. There are two essentials results that will be proposed in this paper. In the first result, we show that a graph (n, 1) − F Caterpillar has a SM labeling and a graph Hu,y has a SEMT labeling. In the second result, we will give an application of these labeling to increase the security level of Affine Cipher in which to encrypt a text on socials media.

Harmonious labeling of the union of m-cycle and n-path graph

A graph G is called harmonious graph if there is an injective mapping f from the set of vertices of G to the set Zq so that the mapping induced a bijection mapping f ∗ from the set of edges of G to Zq , which is defined as f*(xy) = f(x)+ f(y) for each edge xy∈ E(G). The mapping f is called harmonious labeling. Harmonious labeling concept has been developed since 1980. At that time, harmonious labeling is inspired by channel assignment problems. Nowadays, harmonious labeling has been known for various classes of graphs. However, there are still many certain classes of graphs that have not been determined yet whether they can be harmonious labeled or not. One of the class of graphs that have not been determined yet whether it can be harminous labeled or not is graph Cm ∪ Pn In this paper, we prove that there is a harmonious labeling of graph Cm ∪ Pn for some values m and n.

A survey and a new class of graceful unicylic graphs

A graph G admits a graceful labeling if there is a one-to-one map f from the set of vertices of G to such that when an edge xy is assigned the label the resulting set of edge labels is When such a labeling exists, G is called graceful. Rosa showed that a cycle Cn () is graceful if and only if n is congruent to 0 or 3 modulo 4. Truszczyński conjectured that unicyclic graphs, except the cycles in Rosa’s result are graceful. Several classes of unicyclic graphs are known to be graceful. In this article, we present a survey of results related to Truszczyński’s conjecture. We also present a new class of graceful unicyclic graphs.

Palindromic products

A graph G on n vertices is palindromic if there is a vertex-labeling bijection f:V(G) --> {1,2, ... ,n} with the property that for any edge  vw of G, there is an edge xy of G for which f(x)=n-f(v)+1 and f(y)=n-f(w)+1. This notion was defied and explored in a recent paper  [R. Beeler, Palindromic graphs, Bulletin of the ICA, 85 (2019) 85--100]. The paper gives sufficient conditions on the factors of a Cartesian product of graphs that ensure the product is palindromic, but states that it is unknown whether the conditions are necessary. We prove that the conditions are indeed necessary. Further, we prove a parallel result for the strong product of graphs.

3-VERTEX FULL BALANCE INDEX SET OF GRAPHS

Let G beagraphwithvertexset V (G) andedgeset X(G). Consider the set A = {0,1,2}. A labeling f : V (G) → A induces a partial edge labeling f∗ : X(G) → A defined by f∗(xy) = f(x), if and only if f(x) = f(y), for each edge xy ∈ X(G). For i ∈ A, let vf(i) = |{v ∈ V (G) : f(v) = i}| and ef∗(i) = |{e ∈ X(G) : f∗(e) = i}|. Alabelingf ofagraphGissaidtobe3-vertexfriendly if|vf(i)−vf(j)|≤ 1, for all i ∈{0,1,2}. The 3-vertex full balance index set of a graph G is denoted by FBI3v(G) and is defined as{ef∗(i)−ef∗(j),for i,j = 0,1,2 : f∗ runs over all 3-vertex friendly labeling f of G}. In paper, we study 3vertex full balance index set and 3-vertex balance index set of some families of graph.

Further results on Parity Combination Cordial Labeling

Let G be a (p, q)-graph. Let f be an injective mapping from V(G) to {1, 2, …, p}. For each edge xy, assign the label left(genfrac{}{}{0pt}{}{x}{y}right) or left(genfrac{}{}{0pt}{}{y}{x}right) according as x > y or y > x. Call f a parity combination cordial labeling if |ef(0) − ef(1)| ≤ 1, where ef(0) and ef(1) denote the number of edges labeled with an even number and an odd number, respectively. In this paper we make a survey on all graphs of order at most six and find out whether they satisfy a parity combination cordial labeling or not and get an upper bound for the number of edges q of any graph to satisfy this condition and describe the parity combination cordial labeling for two families of graphs.

A comprehensive survey on prime cordial and divisor cordial labeling of graphs

This article presents a short and concise survey on prime cordial and divisor cordial labeling of graphs. A prime cordial labeling of a graph G(V,E) is a bijective function f:V(G) → {1,2,…,|V|} such that if each edge xy is assigned the label 1 if gcd(f(x),f(y)) = 1 and 0 if gcd(f(x),f(y)) > 1, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. Further, a divisor cordial labeling of G is a bijection g: V(G) → {1,2,…,|V|} such that an edge st is assigned the label 1 if one g(s) or g(t) divides the other and 0 otherwise, then the number of edges labeled with 0 and the number of edges labelled with 1 differ by at most 1. We call G a divisor cordial graph if it admits a divisor cordial labeling. This article stands divided into five sections. The first and fifth sections are reserved respectively for introduction and some important references. The second section deals with the prime cordial labeling of certain classes of graphs wherein some important known results have been recalled. The third section deals with the divisor cordial labeling of graphs in which a few known results of high interest have been outlined. In the fourth section we highlight certain conjectures and open problems in respect of the above mentioned labelling that still remain unsolved.

The local edge metric dimension of graph

In this paper, we introduce a new notion of graph theory study, namely a local edge metric dimension. It is a natural extension of metric dimension concept. dG (e,v) = min{d(x,v),d(y,v)} is the distance between the vertex v and the edge xy in graph G. A non empty set is an edge metric generator for G if for any two edges there is a vertex such that . The minimum cardinality of edge metric generator for G is called as edge metric dimension of G, denoted by dimE (G). The local edge metric dimension of G, denoted by dimE (G), is a local edge metric generator of G if for every pair xk,ky of adjacent edges of G. Our concern in this paper is investigating some results of local edge metric dimension on some graphs.

Odd harmonious labeling on squid graph and double squid graph

An injective function f from set of vertices in graph G to a set of {0,1,…,|E| − 1} is called an odd harmonious labeling if the function f induced the edge function f* from the set of edges of G to a set of odd positive integer number {1,3,5,…,2|E| − 1} with f*(xy) = f(x) + f(y) for every edge xy in E. Graph that has an odd harmonious labeling is called odd harmonious graph. The squid graph Tn,k is a graph which is obtained from a cycle Cn and we add k pendant to one vertex of the cycle. It is known that Cn is an odd harmonious graph if and only if n = 0 mod 4. However, by adding at least one pendant in the cycle graph, we can label the new graph odd harmoniously for all even number of vertices. In this paper, we showed that the graph Tn,k and T 2n,k are an odd harmonious graph, for n = 0 (mod 2), n ≥ 4 and k ≥ 1. The construction of the odd harmonious labeling of the graph Tn,k and T 2n,k are inspired by the odd harmonious labeling of Cn for n = 0(mod 4).

Perfect matchings in shadow colorings

This birthday note gives a “non-asymptotic” version of our earlier result with G. N. Sarkozy and Szemeredi [3], in which Endre had the lion’s share. A hypergraph H with vertex set V defines the shadow graph G(H) whose vertex set is V and whose edge set is the set of pairs of V that are covered by some hyperedge of H. An edge coloring C of H defines a multicoloring, the shadow coloring $$C^{\prime}$$ on G(H), by assigning all colors of C to an edge xy of G(H) that appear on some edge of H containing $$\{x, y\}$$ . A matching in a graph is a set of pairwise disjoint edges. A matching in a graph is perfect if it covers all vertices of the graph. I show that in every (r −1)-coloring C of the complete r-uniform hypergraph $$K^{r}_{n}$$ there is a monochromatic perfect matching in the shadow coloring $$C^{\prime}$$ (assumingn ≥ r ≥ 2 and n is even).

Edge Irregular Reflexive Labeling on Corona of Path and Other Graphs

Let G(V, E) be an undirected and simple graph with vertex set V and edge set E Define a k -labeling f on G such that the element belong to E are labeled with integers {1,2,…,ke } and the element belong to V are labeled with even integers {0,2,…,2kv }, where k = max{ke ,2kv }. A k -labeling f is mentioned as an edge irregular reflexive k -labeling if distinct edges have distinct weight. The weight of edge xy is denoted by wt(xy) and defined as wt(xy) = f (x) + f (xy) + f (y). A minimum k for which G has an edge irregular reflexive k -labeling is called reflexive edge strength of G and denoted by res(G). This paper contains investigation of edge irregular reflexive k -labeling for corona of path and other graphs and determination of their reflexive edge strengths.

New Family of Parity Combination Cordial Labeling of Graph

Let G be a (p, q) graph. Let f be an injective map from V(G) to {1, 2, …, p}. For each edge xy, assign the label         y x or         x y according as x > y or y > x. f is called a parity combination cordial labeling (PCC-labeling) if f is a one to one map and | ef(0) − ef(1) |  1 where ef(0) and ef(1) denote the number of edges labeled with an even number and odd number respectively. A graph with a parity combination cordial labeling is called a parity combination cordial graph (PCC-graph). In this paper we investigate the PCC- labeling of the graph G , It is obtained by identifying a vertex vk in G and a vertex of degree n in Hn, where G is a PCC graph with p vertices and q edges under f with f(vk) = 1.

Group 3 S Cordial Remainder Labeling

Let G  (V(G),E(G)) be a graph and let 3 g :V(G)S be a function. For each edge xy , assign the label r where r is the remainder when og(x) is divided by og(y) or og(y) is divided by og(x) according as og(x)  o g( y) or og( y)  o g(x) . The function g is called a group 3 S cordial remainder labeling of G if | ( ) ( ) | 1 g g v x  v y  and | (0) (1) | 1 g g e  e  , where ( ) g v x denotes the number of vertices labeled with x and ( ) g e i denotes the number of edges labeled with i (i  0,1) . A graph G which admits a group 3 S cordial remainder labeling is called a group 3 S cordial remainder graph. In this paper, we introduce the concept of group 3 S cordial remainder labeling. We prove that some standard graphs admit a group 3 S cordial remainder labeling.

Beck’s Zero-Divisor Graph in the Realm of Signed Graph

For a commutative ring R with unity ($$1\ne 0$$), the zero-divisor graph of R, denoted by $$\varGamma (R)$$, is a simple graph with vertices as elements of R and two distinct vertices are adjacent whenever the product of the vertices is zero. Further, its signed zero-divisor graph is an ordered pair $$\varGamma _{\varSigma }(R):= (\varGamma (R), \sigma )$$, where for an edge xy, $$\sigma (xy)$$ is ‘$$+$$’ if either x or y or both is a nonzero zero-divisor and ‘−’ otherwise. This article aims at gaining a deeper insight into signed zero-divisor graphs by investigating properties such as balancing, clusterability, sign-compatibility, consistency, $$\mathcal {C}$$-sign-compatibility, and $$\mathcal {C}$$-consistency.

Total Edge Irregularity Strength of Arithmetic Book Graphs

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 .

Further results on 3-product cordial labeling

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.

On the [formula omitted]-choosability of planar graphs without adjacent cycles of length [formula omitted

A (k,d)-list assignment L of a graph G is a mapping that assigns to each vertex v a list L(v) of at least k colors satisfying |L(x)∩L(y)|≤d for each edge xy. A graph G is (k,d)-choosable if there exists an L-coloring of G for every (k,d)-list assignment L. This concept is also known as choosability with separation. In this paper, we prove that any planar graph G is (3,1)-choosable if any i-cycle is not adjacent to a j-cycle, where 5≤i≤6 and 5≤j≤7.

3-product cordial labeling of some snake graphs

Let G be a (p,q) graph. A mapping f : V (G) → {0, 1, 2} is called 3-product cordial labeling if |v????(i) − v???? (j)| ≤ 1 and |e???? (i) − e???? (j)| ≤ 1 for any i, j ∈ {0, 1, 2},where v???? (i) denotes the number of vertices labeled with i, e???? (i) denotes the number of edges xy with ????(x)????(y) ≡ i(mod3). A graph with 3-product cordial labeling is called 3-product cordial graph. In this paper we investigate the 3-product cordial behavior of alternate triangular snake, double alternate triangular snake and triangular snake graphs.

Colouring of generalized signed triangle-free planar graphs

We view an undirected graph G as a symmetric digraph, where each edge xy is replaced by two opposite arcs e=(x,y) and e−1=(y,x). Assume S is an inverse closed subset of permutations of positive integers. We say G is S-k-colourable if for any mapping σ:E(G)→S with σ(x,y)=(σ(y,x))−1, there is a mapping f:V(G)→[k]={1,2,…,k} such that σe(f(x))≠f(y) for each arc e=(x,y). The concept of S-k-colourable is a common generalization of several other colouring concepts. This paper is focused on finding the sets S such that every triangle-free planar graph is S-3-colourable. Such a set S is called TFP-good. Grötzsch’s theorem is equivalent to say that S={id} is TFP-good. We prove that for any inverse closed subset S of S3 which is not isomorphic to {id,(12)}, S is TFP-good if and only if either S={id} or there exists a∈[3] such that for each π∈S, π(a)≠a. It remains an open question to determine whether or not S={id,(12)} is TFP-good.