Further results on 3-product cordial labeling

We study the Open image in new window (shortly, Open image in new window ) problem on geometric intersection graphs such as interval, circular-arc, permutation, unit-disk, outer-string graphs, etc. Given a Open image in new window graph \(G=(V,E)\), where each vertex in V is colored with either “ Open image in new window ” or “ Open image in new window ”, the BCS problem seeks a maximum cardinality induced connected subgraph H of G such that H is Open image in new window , i.e., H contains an equal number of red and blue vertices. We study the computational complexity landscape of the BCS problem while considering geometric intersection graphs. On one hand, we prove that the BCS problem is NP-hard on the unit disk, outer-string, complete grid, and unit square graphs. On the other hand, we design polynomial-time algorithms for the BCS problem on interval, circular-arc and permutation graphs. In particular, we give algorithms for the Open image in new window problem on both interval and circular-arc graphs, and those algorithms are used as subroutines for solving the BCS problem on the same classes of graphs. Finally, we present a FPT algorithm for the BCS problem on general graphs.

Partitioning subclasses of chordal graphs with few deletions

On polygon numbers of circle graphs and distance hereditary graphs

Let G(V, E) be a finite, simple and undirected graph with a vertex set V and an edge set E. An edge irregular total k-labeling is a map f : V ⋃ E → {1, 2, …, k} such that for any two different edges xy and x′y′ in E, ω(xy) ≠ ω(x′y′) where ω(xy) = f(x) + f(y) + f(xy). The minimum k for which the graph G admits an edge irregular total k-labeling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of book graphs and double book graphs. In this paper, we construct an edge irregular total k-labeling that can be used for book graphs, double book graphs, and triple book graphs. We also show the exact value of the total edge irregularity strength of triple book graphs.

In this paper we used the cubic roots of unit group together with the concept of cordiality in graph theory to introduce a new method of labeling, this construed cubic cordial labeling can be applied to all paths, cycles, fans and wheel graphs. Moreover, some other properties are investigated and show that the union of any two cycles and the union any two paths are cubic cordial graphs. Also, we study the cubic cordiality for the union of any cycle with a path.

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.

Cordial labeling of hypertrees

A k-edge product cordial labelling is a variant of the well-known cordial labelling. In this paper, a balanced k-edge product cordial labelling is suggested and some sufficient conditions for balanced 3-edge product cordial graphs are proved. Moreover, a construction of graphs admitting a balanced 3-edge product cordial labelling is presented.

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,E) be a graph with p vertices and q edges. A Extended Mean Cordial Labeling of a Graph G with vertex set V is a bijection from V to {0, 1,2} such that each edge uv is assigned the label where ⌈ x ⌉ is the least integer greater than or equal to x with the condition that 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 almost 1. The graph that admits an Extended Mean Cordial Labeling is called Extended Mean Cordial Graph. In this paper, we proved that tree related graphs Hdn, K 1,n, Tgn, <K1,n:n> are Extended Mean Cordial Graphs.

In 2012, Ponraj et al. defined k-product cordial labeling as follows: Let [Formula: see text] be a map from [Formula: see text] to [Formula: see text] where [Formula: see text] is an integer, [Formula: see text]. For each edge [Formula: see text] assign the label [Formula: see text] [Formula: see text]. [Formula: see text] is called a k-product cordial labeling if [Formula: see text], and [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] denote the number of vertices and edges, respectively, labeled with [Formula: see text] [Formula: see text]. A graph that admits k-product cordial labeling is called k-product cordial graph. Later, we proved that several families of graphs are k-product cordial graphs. In this paper, we show that the product of graphs admit k-product cordial labeling.

A mapping f:V(G)→{0,1,2} is called a 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 a 3-product cordial labeling is called a 3-product cordial graph. In this paper, we investigate the 3-product cordial behaviour for some standard graphs.

In this paper, we introduce a new labeling called Totally Vertex-Magic Cordial(TVMC) labeling. A graph G(p,q) is said to be TVMC with a constant C if there is a mapping f:V(G)∪E(G)→{0,1} such that [ f(a)+∑b∈N(a)f(ab) ]≡C (mod 2) for all vertices a∈V(G) and |nf(0)−nf(1)| ≤1, where N(a) is the set of vertices adjacent to the vertex a and nf(i)(i=0,1) is the sum of the number of vertices and edges with label i.

In this paper, we investigate some new families of edge sum divisor cordial graphs, face edge sum divisor cordial graphs and total face edge sum divisor cordial graphs.Keywords -Edge sum divisor cordial labelling, Edge sum divisor cordial graph, face edge sum divisor cordial graph, total face edge sum divisor cordial graph, switching of any vertex.

A bijection f: V(G) → {1,2,··· ,|V(G)|} induces an edge labeling f∗: E(G) → {0,1} such that for any edge uv in G, f∗(uv) = 1 if gcd(S,D) = 1 and f∗(uv) = 0 otherwise, where S = f(u)+ f(v) and D = |f(u)− f(v)|. The labeling f is called SD-prime cordial labeling if |ef∗(0) − ef∗(1)|≤1. We say that G is SD-prime cordial graph if it admits SD-prime cordial labeling. In this paper, we prove that certain classes of zero-divisor graphs of commutative rings are SD-prime cordial graphs.