On the common neighborhood graphs

2021

The near common neighborhood graph of a graph G, denoted byncn(G) is defined as the graph on the same vertices of G, two verticesare adjacent in ncn(G), if there is at least one vertex in G not adjacentto both the vertices. In this research paper, the conditions for ncn(G)to be disconnected are discussed and also we present characterizationfor graph ncn(G) to be hamiltonian and eulerian

Electronic Notes in Discrete Mathematics, 2009

A set S ⊆ V is a neighborhood set of G, if G = v∈S N [v] , where N [v] is the sub graph of G induced by v and all vertices adjacent to v. The neighborhood number η(G) of G is the minimum cardinality of a neighborhood set of G. In this paper, we extended the concept of neighborhood number and its relationship with other related parameters are explored.

Doklady Mathematics, 2010

During the last few years, my research has been in graph theory and has led to several publications and pre-publications . Firstly I will introduce the results of and .

2008

A local coloring of a graph G is a function c : V (G) −→ N having the property that for each set S ⊆ V (G) with 2 ≤ |S| ≤ 3, there exist vertices u, v ∈ S such that |c(u) − c(v)| ≥ mS , where mS is the size of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ (c). The local chromatic number of G is χ (G) = min{χ (c)}, where the minimum is taken over all local colorings c of G. If χ (c) = χ (G), then c is called a minimum local coloring of G. The local coloring of graphs introduced by Chartrand et. al. in 2003. In this paper, following the study of this concept, first an upper bound for χ (G) where G is not complete graphs K4 and K5, is provided in terms of maximum degree Δ(G). Then the exact value of χ (G) for some special graphs G such as the cartesian product of cycles, paths and complete graphs is determined.

Journal of Combinatorial Theory, Series B, 1974

Journal of Combinatorial Theory, 1989

We investigate the relationship between the cardinality of the union of the neighborhoods of an arbitrary pair of nonadjacent vertices and various hamiltonian type properties in graphs. In particular, we show that if G is 2-connected, of order p ≥ 3 and if for every pair of nonadjacent vertices x and y: (a) , then G is traceable,(b) , then G is hamiltonian, and if G is 3-connected and(c) , then G is hamiltonian-connected.

Journal of Graph Theory, 2006

Dirac proved that a graph G is hamiltonian if the minimum degree δ(G) ≥ n/2, where n is the order of G. Let G be a graph and A ⊆ V(G). The neighborhood of A is N(A) = {b : ab ∈ E(G) for some a ∈ A}. For any positive integer k, we show that every (2k − 1)-connected graph of order n ≥ 16k 3 is hamiltonian if |N(A)| ≥ n/2 for every independent vertex set A of k vertices. The result contains a few known results as special cases. The case of k = 1 is the classic result of Dirac when n is large and the case of k = 2 is a result of Broersma, Van den Heuvel, and Veldman when n is large. For general k, this result improves a result of Chen and Liu. The lower bound 2k − 1 on connectivity is best possible in general while the lower bound 16k 3 for n is conjectured to be unnecessary.

Discrete Mathematics, 1986

The notion of neighborhood perfect graphs is introduced here as follows. Let G be a graph, ~N(G) denote the maximum number of edges such that no two of them belong to the same subgraph of G induced by the (closed) neighborhood of some vertex; let PN(G) be the minimum number of vertices whose neighborhood subgraphs cover the edge set of G. Then G is called neighborhood perfect if PN(G') = CrN(G' ) holds for every induced subgraph G' of G. It is expected that neighborhood perfect graphs are perfect also in the sense of Berge. We characterize here those chordal graphs which are neighborhood perfect. In addition, an algorithm to compute PN(G) = ON(G) is given for interval graphs.

2013

We give a new recursive method to compute the number of cliques and cycles of a graph‎. ‎This method is related‎, ‎respectively to the number of disjoint cliques in the complement graph and to the sum of permanent function over all principal minors of the adjacency matrix of the graph‎. ‎In particular‎, ‎let $G$ be a graph and let $overline {G}$ be its complement‎, ‎then given the chromatic polynomial of $overline {G}$‎, ‎we give a recursive method to compute the number of cliques of $G$‎. ‎Also given the adjacency matrix $A$ of $G$ we give a recursive method to compute the number of cycles by computing the sum of permanent function of the principal minors of $A$‎. ‎In both cases we confront to a new computable parameter which is defined as the number of disjoint cliques in $G$‎.

A graph G is said to be cyclable if for each orientation → G of G, there exists a set S of vertices such that reversing all the arcs of → G with one end in S results in a hamiltonian digraph. Let G be a 3-connected graph of order n ¿ 36. In this paper, we show that if for any three independent vertices x1, x2 and x3, |N (x1) ∪ N (x2)| + |N (x2) ∪ N (x3)| + |N (x3) ∪ N (x1)| ¿ 2n + 1, then G is cyclable. ?

2021

An edge labeling of a graph G = (V, E) is said to be local antimagic if it is a bijection f:E →{1,… ,|E|} such that for any pair of adjacent vertices x and y, f^+(x)≠ f^+(y), where the induced vertex label of x is f^+(x)= ∑_e∈ E(x) f(e) (E(x) is the set of edges incident to x). The local antimagic chromatic number of G, denoted by χ_la(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, several sufficient conditions to determine the local antimagic chromatic number of the join of graphs are obtained. We then determine the exact value of the local antimagic chromatic number of many join graphs.

A subset S of a vertex set V(G) such that a vertex in V(G) -S is adjacent to a vertex in S is a dominating set of G. The minimum cardinality among all dominating sets of the graph is the domination number. A subset S of the vertex set is a neighbourhood set if the graph is the union of the sub graphs induced by the closed neighbourhoods of the vertices of the subset S. The minimum cardinality among all minimal neighbourhood sets is the neighbourhood number of the graph. The notion of a dominating set of a graph was introduced by Oystein Ore in 1962. This concept is generalized to the Neighbourhood set by Sampathkumar and Neeralagi in 1985.

AKCE International Journal of Graphs and Combinatorics

The matching number of a graph G is the size of a maximum matching in the graph. In this note, we present a sufficient condition involving the matching number for the Hamiltonicity of graphs.

Discussiones Mathematicae Graph Theory, 2009

For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G, the neighborhood color set NC(v) is the set of colors of the neighbors of v. The coloring c is called a set coloring if NC(u) = NC(v) for every pair u, v of adjacent vertices of G. The minimum number of colors required of such a coloring is called the set chromatic number χ s (G) of G. The set chromatic numbers of some well-known classes of graphs are determined and several bounds are established for the set chromatic number of a graph in terms of other graphical parameters.