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Boissonnat, P. Chenin, A. Cohen, C. Gout, T. Lyche, and M. Buzaglo, R. Pinchasi, and G. Topological hypergraphs. Kyncl, V. Stolar, and P. Universal sets for straight-line embeddings of bicolored graphs. To appear. Hurtado and C. Plane geometric graph augmentation: a generic perspective. Pach, D. Survey on decomposition of multiple coverings. Refereed articles in conference proceedings B.
Aichholzer, S. Ramos, and G. In 28th Ann. ACM Symp. Abellanas, A. Bajuelos, S. Canales, M. Connecting red cells in a bicolour voronoi diagram. Aurenhammer, B. Triangulations with circular arcs. Allen, G. Aloupis, L. Barba, P. Sum of squared edges for mst of a point set in a unit square.
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Meijer, and Y. Universal point subsets for planar graphs. Arkin, J. Kumar, J. Mitchell, B. Palop, P. Bichromatic 2-center of pairs of points. Asinowski, J. Cardinal, N. Cohen, S. Collette, T. Hoffmann, K. Lason, P. Micek, G. Rote, and T. Coloring hypergraphs induced by dynamic point sets and bottomless rectangles.
Asinowski, T. Miltzow, and G. Quasi-parallel segments and characterization of unique bichromatic matchings extended abstract. Korman, Y. Okamoto, and H. Barba, S. Durocher, R. Fraser, F. Mehrabi, D. Mondal, J. Morrison, M.
Skala, and M. On k-enclosing objects in a coloured point set. Flores, S. Langerman, and G. Circle separability queries in logarithmic time. In CCCG , pages , Silveira, and K. Space-time trade-offs for stack-based algorithms. Rote, and M. Fabila-Monroy, P. Sakai, J. Urrutia, and I. On balanced 4-holes in bichromatic point sets. A tight colored Tverberg theorem for maps to manifolds extended abstract. Korman, and P. In ESA , pages , Hoffmann, S. Coloring dynamic point sets on a line. Ito, and S. Cardinal, K.
Knauer, P. Micek, and T. Making octants colorful and related covering decomposition problems. Claverol, D. Garijo, M. Korman, C. Stabbing segments with rectilinear objects. On planar point sets with the pentagon property. Cibulka, P. Gao, M. Valla, and P. Polynomial bounds on geometric ramsey numbers of ladder graphs. Witness bar visibility graphs. Fabila-Monroy, and P. On the number of radial orderings of colored planar point sets. The 1-Center and 1-Highway problem. Locating a service facility and a rapid transit line. Locating a service facility and a rapid transit line of variable length.
Dorzan, G. Leguizamon, and E. Metaheuristic approaches for the minimum dilation triangulation problem. Fabila-Monroy and C. Covering islands in plane point sets. Fabila-Monroy, C. Huemer, and E. The expected number of points in circles. Note on the number of obtuse angles in point sets. Felsner, M. Kaufmann, and P. The graphs that can be drawn orthogonally with one bend per edge. Frati, J. Gudmundsson, and E. On the number of upward planar orientations of maximal planar graphs.
A note on the number of empty triangles. Tejel, and P. AMS : 47H10, 54H Introduction Fixed point theory plays a very crucial role in the development of nonlinear analysis. The Banach  fixed point theorem for contraction mapping has been generalized and extended in many directions. This famous theorem can be stated as follows. The Banach contraction principle with rational expressions have been expanded and some fixed point and common fixed point theorems have been obtained in [4, 5].
Recently, Azam et al. Complex-valued metric space is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, mechanical engineering and electrical engineering, for more details, see, [7, 8]. In this paper, we establish common fixed point results for generalized contraction involving rational expression in the framework of complex valued metric spaces.
Note that 0. The following definition was introduced by Azam et al. Then d is called a complex valued metric on X and X, d is called a complex valued metric space. Then X, d is a complex valued metric space. If every Cauchy sequence converges in X, then X is called a complete complex valued metric space. Main Results In this section we shall prove some common fixed point results under generalized contraction involving rational expression in the framework of complex valued metric spaces.
Saluja and T have a unique common fixed point in X. Then from 3. This shows that w is a common fixed point of S and T. We now show that S and T have a unique common fixed point. This shows that S and T have a unique common fixed point in X. This completes the proof. Corollary 3. Then T has a unique fixed point in X. This shows that T has a unique fixed point in X. Proof In view of Theorem 3.
Now we are required to show that g is a common fixed point of all the components maps of both the families. Saluja Corollary 3. Then F and G have a unique common fixed point in X. Proof By Corollary 3. The rest of the proof is same as that of Corollary 3. The following example demonstrates the superiority of Bryant see,  theorem over Ba- nach contraction theorem. Finally, we conclude this paper with an illustrative example which satisfied all the condi- tions of Corollary 3.
Saluja course 0 is the unique fixed point of T. Conclusion In this paper, we establish common fixed point theorems using generalized contraction involving rational expression in the setting of complex-valued metric spaces and give an example in support of our result. Our results extend and generalize several results from the current existing literature. References  A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Banach, Surles operation dans les ensembles abstraits et leur application aux equation integrals, Fund.
Bryant, A remark on a fixed point theorem for iterated mappings, Amer. Monthly, 75 , Fisher, Common fixed points and constant mapping satisfying rational inequality, Math. Notes Univ. Kobe , Khan, Fixed points, common fixed points and constant mappings, Studia Sci. Imdad, J. Ali and M. Tanveer, Coincedence and common fixed point theorem for nonlinear contractions in Menger PM spaces, Chaos Solitones Fractals, 42 , Sintunavarat and P.
Kumam, Generalized common fixed point theorem in complex valued metric spaces with applications, J. Verma and H. Pathak, Common fixed point theorem using property E. A in complex valued metric spaces, Thai J. Riyas and K. Geetha Department of Mathematics, T. But if n is prime, it is always Hamiltonian. Key Words: Smarandache-Cayley graph, Cayley graph, dihedral group, Hamiltonian cycle, complete graph. AMS : 05C Arthur Cayley introduced the Cayley graphs of groups and it has received much attention in the literature. Brian Alspach et al.
Main Results In this section we deals with some basic definitions and terminologies of group theory and graph theory which are needed in sequel. For details see Fraleigh , Gallian and Diestel Geetha Definition 2. The centralizer of an element x in G , CG x is the set of all element in G that commute with x. Generally all reflec- tions are involutions and rotations may or may not. If n is odd, e is the only involution in G1 and G2 consist of reflections along perpendicular bisector of sides only.
Except for e , generally n G1 and G2 never commute and G2 is non-abelian , but if n is even, g 2 is the only involution in G1 which commute G2. A Hamiltonian cycle is a closed Hamiltonian path. A graph V, E is said to be Hamiltonian, if it contains a Hamiltonian cycle. A complete graph of n vertices is denoted as Kn. Geetha Let us consider the following cases. Corollary 2. Thus the induced subgraph with vertex set CG x of the Cayley graph Cay G, x is a bipartite graph on four vertices.
Then by Corollary 2. As in the proof Theorem 2. Then by Theorem 2. Also we have ,by Theorem 2. AMS : 53A40, 53B Introduction Mannheim curve was firstly defined by A. Mannheim in As a result they called these new curves as Mannheim partner curves. For more detail see in . Shukla, O. Prasad and Bindu Kumari  and C.
The Homogeneity of f in 1. Differentiating 1. The concept of concurrent vector field has been given by Matsumoto and K. Eguchi  and S. It has been proved by by Matsumoto that bi and its contravariant components bi are functions of coordinates alone. Therefore from 1. The successive differentiation of 1. From 2. Differentiating 2.
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Using equation 3. So equation 3. Therefore we have the following result.
In the theorem 3. Thus, we have the following result. The condition 3. Hence equation 3. Hence we have the following result. In view of these equations, we have from 4. In view of 4. Pandey and Khageshwar Manda Contracting 4. L From 2. Therefore from 5. Kyoto Univ. Matsumoto and K. Eguchi, Finsler space admitting a concurrent vector field, Tensor N. Tachibana, On Finsler spaces which admit a concurrent vector field, Tensor N. Narayankar and Lokesh S. The peripheral distance energy of a graph G is the sum of the absolute values of the eigenvalues of Dp -matrix of G.
The sum of the distances between all pairs of peripheral vertices is a peripheral Wiener index of a graph G. In this paper, we study some preliminary facts of Dp -matrix of G and give some bounds for peripheral distance energy of a graph G. Specially the bounds are presented for a graph of diameter less than 3. Key Words: Distance, peripheral Wiener index, peripheral distance matrix, peripheral distance energy. AMS : 05C12, 05C Let u and v be two vertices of a graph G.
The distance d u, v G between the vertices u and v is the length of a shortest path connecting u and v. The eccentricity e v of a vertex v in a graph G is the distance between v and a vertex farthest from v in G. The diameter diam G of G is the maximum eccentricity of G, while the radius rad G is the smallest eccentricity of G. The set of peripheral vertices of G is called as periphery and is denoted as P G. We claim that the adjacency matrix of a graph is the distance based matrix such that the entries of adjacency matrix are 1 if the distance between two vertices is 1 and 0 otherwise.
For the application and the background of the distance matrix on the chemistry, one can refer to [1, 32].
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Peripheral Distance Energy of Graphs 89 [dij ], where dij is the distance between two peripheral vertices vi and vj in G. The peripheral distance energy Dp -energy in short of a graph G is defined as the sum of the absolute values of Dp - eigenvalues of Dp -matrix of G. Observe that the graph energy E G in past a few years has been extensively studied and surveyed in Mathematics and Chemistry [8, 11, 14, 18, 19, 20, 21, 22, 25, 26, 27, 29, 30, 31, 33]. The eigenvalues of Dp G are said to be the peripheral distance eigenvalues or Dp -eigenvalues in short of G.
Hence EDp -energy of G is This paper is organized as follows: In the forthcoming section some preliminary facts of peripheral distance matrix Dp G of G are obtained. In section 3 bounds of peripheral distance energy in terms peripheral Wiener index are deduced. In section 4 bounds for the peripheral distance energy are established. In the last section the smallest peripheral distance energy of a graph is obtained thereby posing an open problem for the maximum peripheral distance energy.
Preliminary Results Lemma 2. If we partition the vertex set V G of a graph into two sets, with peripheral vertices in one set and non-peripheral vertices in other. Then the sum of the distances between all pairs of peripheral vertices is the peripheral Wiener index of a graph G.
Proof From the Lemma 2. Bounds for the Peripheral Distance Energy Theorem 4. Proof We know that, for non-negative numbers the arithmetic mean is not smaller than the geometric mean. To prove the upper bound we follow the ideas of Koolen and Moulton [18, 19], who obtained an analogous upper bound for ordinary graph energy E G. Proof The proof follows from Theorem 4. The Smallest Peripheral Distance Energy of a Graph By studying the bounds for peripheral distance energy, there arise a common question that, which n vertex graphs with k peripheral vertices have the smallest and greatest peripheral distance energy.
Among all n-vertex connected graphs with k peripheral vertices the complete graph is the unique graph with the smallest peripheral distance energy. Theorem 5. Proof Let G be a graph with k peripheral vertices and Kk be a complete graph on k peripheral vertices. Let A be a peripheral distance matrix of Kk. Hence, we conclude that the peripheral distance energy of a graph with k peripheral vertices is greater than the peripheral distance energy of a complete graph on k vertices.
However, in , the authors have given the direct reason for the proof of the conjecturer in . Balaban, D. Ciubotariu, M. Medeleanu, Topological indices and real number vertex invariants based on graph eigenvalues or eigenvectors, J. Zhou, A. Gutman Eds. Gutman, The energy of a graph: old and new results, in: A. Betten, A. Kohnert, R. Laue, A. Wassermann Eds. Berlin, , pp. Gutman, The energy of a graph, Ber. Forsch-ungszentram Graz. Gutman, Distance in Thorny graph, Publ.
Math Beograd , 63 77 Gutman, B. Furtula, H. Gutman, O. Gutman, M. Medeleanu, On the structure dependence of the largest eigenvalue of the distance matrix of an Alkane, Indian J. Gutman, S. Zare Firoozabadi, J. Hua, M. Wang, Unicyclic graphs with given number of vertices and minimal energy, Lin. Algebra Appl. Indulal, I. Gutman, On the distance spectra of some graphs, Math.
Gutman, A. Narayankar, Lokesh S. Koolen, V. Moulton, Maximal energy graphs, Adv. Moulton, Maximal energy bipartite graphs, Graph. Li, Y. Shi, and I. Nikiforov, The energy of graphs and matrices, J. Nikiforov, Graphs and matrices with maximal energy, J. Ramane, I. Gutman, D. Ramane, D. Revankar, I. Rao, B. Acharya, H. Ramane, H. Walikar, S. Acharya, P. Hampiholi, S. Jog, I. Gutman, Spectra and energies of iterated line graphs of regular graphs, Appl.
Ramane and H. Shparlinski, On the energy of some circulant graphs, Lin. Peripheral Distance Energy of Graphs  S. Jog, R. Yan, L. Ye, On the maximal energy of trees with a given diameter, Appl. Ye, X. Yu, X. Lv, Minimal energy of trees with k pendent vertices, Lin. Zhou, N. Zhou, I. Gutman, J. Rada, and L. Chaubey1 , Arunima Mishra2 and A. Pandey3 1. AMS : 53B40, 53C Since a concurrent vector field is a func- tion of x i. Singh and Prasad [14,11] generalized the concept of concurrent vector field and introduced the semi-parallel and concircular vector fields which are functions of x only.
Such a Finsler metric was first introduced by G. Randers from the standpoint of general theory of relativity and applied to the theory of the electron microscope by R. Society, 69 , Society, , Bejancu and N. Papaghiuc, Semi invariant submanifolds of a Sasakain manifold, Al. Cuza, lasi, Sect I a Math. Reidel Publishing Co. Chen, Riemannian submnaifolds, in Handbook of differential geometry, Vol.
Dillen and L. Verstraelen, North-Holland, Amsterdam, , pp. Chen, S-invariants inequalities of submanifolds and their applications, in : Topics in differential geometry, Ed. Bucharest, , pp. Debrecen, 66 , Kenmotsu, A class of almost contact Riemannian maifolds, Tohoku Math. Papaghiuc, Semi-invariant submanifolds in a Kenmotsu manifolds, Rend. Univ, Al. Cuza lasi Sect. I a Mat. Raheem, A. Abstract: The concept of labeling has its origin in the works of Stewart , Kotzig and Rosa Later on Enomoto, Llado, Nakamigawa and Ringel defined a super a, 0 -edge-antimagic total labeling and proposed the conjecture that every tree is a super a, 0 -edge-antimagic total graph.
In the favour of this conjecture, the present paper deals with different results on antimagicness of a class of trees, which is called subdivided stars. Key Words: Smarandachely super a, d -edge-antimagic total labeling, super a, d -edgeantimagic total labeling, stars and subdivision of stars. Introduction All graphs in this paper are finite, undirected and simple. A general reference for graph-theoretic ideas can be seen in . A labeling or valuation of a graph is a map that carries graph elements to numbers usually to positive or non-negative integers.
In this paper, the domain will be the set of all vertices and edges and such a labeling is called a total labeling. Some labelings use the vertex-set only or the edge-set only and we shall call them vertex-labelings or edge-labelings, respectively. In Definitions 1. The definition of an a, d -EAT labeling was introduced by Simanjuntak, Bertault and Miller in  as a natural extension of magic valuation defined by Kotzig and Rosa .
A super a, d -EAT labeling is a natural extension of the notion of super edge-magic labeling defined by Enomoto, Llado, Nakamigawa and Ringel. Moreover, Enomoto et al. Conjecture 1. In the favor of this conjecture, many authors have considered a super a, 0 -EAT labeling for different particular classes of trees.
Lee and Shah  verified this conjecture by a computer search for trees with at most 17 vertices. For different values of d, the results related to a super a, d -EAT labeling can be found for w-trees , stars , subdivided stars [14, 15, 21, 22, 29, 30], path-like trees , caterpillars [17, 18, 25], disjoint union of stars and books  and wheels, fans and friendship graphs , paths and cycles  and complete bipartite graphs .
For detail studies of a super a, d -EAT labeling reader can see [2, 4, 5, 7, ]. Ngurah et al. Salman et al. Moreover, Javaid et al. Basic Results In this section, we present some basic results which will be used frequently in the main results. The lower and upper bounds of the minimum edge-weight a for another subclass of subdivided stats established by Salman et al. Moreover, the following lemma presents the lower and upper bound of the minimum edgeweight a for the most generalized subclass of subdivided stars proved by Javaid and Akhlaq: 1 Lemma 2.
Let a v, e -graph G be a super a, d -EAT. Let us consider the following proposition which we will use frequently in the main results. Proposition 2. Therefore, by Proposition 2. Similarly by Proposition 2. Proof Let us consider the vertices and edges of G, as defined in Theorem 3. Acknowledgement The authors are indebted to the referee and the editor-in-chief, Dr. Linfan Mao for their valuable comments to improve the original version of this paper.
Lin, M. Miller and M. Youssef, Edge-antimagic graphs, Discrete Math. Miller and R. Simanjuntak, New constructions of magic and antimagic graph labelings, Utilitas Math. Lin and F. Muntaner-Batle, Super edge-antimagic labelings of the path-like trees, Utilitas Math. Shafiq, A method to generate large classes of edge-antimagic trees, Utilitas Math. Sudarsana and Y. Cholily, How to construct new super edge-magic graphs from some old ones, J. MIHIM , , — Miller, J. Ryan and M. Nakamigawa and G. Ichishima and F.
Muntaner-Batle, The place of super edgemagic labelings among other classes of labelings, Discrete Math. Muntaner-Batle, On super edge-magic graph, Ars Combinatoria, 64 , 81— Hussain, K. Ali and K. Dar, Super edge-magic total labeling on w-trees, Utilitas Math. Ali and H. Shaker, On super edge-magic total labeling on subdivision of trees, Utilitas Math. Bhatti, On super a, d -edge-antimagic total labeling of subdivided stars, Utilitas Math. Bhatti, Super a, d -edge-antimagic total labeling of subdivided stars and w-trees, Utilitas Math.
Rosa, Magic valuations of finite graphs, Canad. Kong, On super edge-magic n stars, J. Simanjuntak and E. Ngurah and N. Izzati, On super edge-magic total labeling of a subdivision of a star Sn , Utilitas Mthematica, 81 , — Bertault and M. Miller, Two new a, d -antimagic graph labelings, Proc. Simanjuntak, Edge-magic total labelings of wheel, fans and friendship graphs, Bull. ICA, 35 , 89— Miller, Slamin and M. Ponraj, S. Abstract: In this paper, we introduce a new type of graph labeling known as total mean cordial labeling. In this paper, we study some classes of graphs and their Total Mean Cordial behaviour.
Key Words: Smarandachely total mean cordial labeling, total mean cordial labeling, path, cycle, wheel, complete graph, complete bipartite graph. Introduction Unless mentioned otherwise, a graph in this paper shall mean a simple finite and undirected. For all terminology and notations in graph theory, we follow Harary . Graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labeling plays an important role of various fields of science and few of them are astronomy, coding theory, x-ray crystallography, radar, circuit design, communication network addressing, database management, secret sharing schemes, and models for constraint programming over finite domains .
The graph labeling problem was introduced by Rosa and he has introduced graceful labeling of graphs  in the year In , Cahit  introduced the cordial labeling of graphs. In , Ponraj et al. Motivated by these labelings, we introduce a new type of labeling, called total mean cordial labeling. In this paper, we investigate the total mean cordial labeling behaviour of some graphs like path, cycle, wheel, complete graph etc. Let x be any real number. Main Results Definition 2. A graph with a total mean cordial labeling is called total mean cordial graph.
Case 1. The following table Table 1 shows that the above vertex labeling f is a total mean cordial labeling. Proof Let Cn : u1 u2. But this is an impossible one. The labeling given in Figure 1 shows that C6 is total mean cordial. The labeling f defined in case 2 of Theorem 2. Case 3. The labeling f defined in case 3 of Theorem 2. Lemma 2. Proof As in Lemma 2. Proof Suppose f is a total mean cordial labeling of Kn.
Proof Suppose there exists a total mean cordial labeling of Kn , say f. Proof Proof follow from Lemmas 2. Case 4. Case 5. Case 6. Subcase 2. While counting the total number of zeros each vertices ui along with the edges uui contributes 2 zeros. This implies evf 0 is an odd number, a contradiction. Subcase 3. Proof Let Pn : u1 u2.
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For T3 , the vertex labeling in Figure 6 is a total mean cordial labeling. References  I. Cahit, Cordial graphs: a weaker version of graceful and harmonious graphs, Ars combin. Harary, Graph theory, Addision wesley, New Delhi, Mohanty and A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs Internat. Ponraj, M. Sivakumar and M. Smarandache Constant Conjecture The constant cS is the smallest solution of equation 1. The notation was adopted in this article. We present some conjectures and theorems regarding the distribution of prime numbers.
Bertrand formulated this theorem in This assumption was proven for the first time by Chebyshev in Finding the smallest interval that contains at least one prime number p, was a very hot topic. Squaring 1. In Firoozbakht verified the inequality 1. Currently the table was completed up to the position 75 [15, 24].
Remark 1. Conjecture Wolf Furthermore, the bounds presented below suggest yet another growth rate, namely, that of the square of the so-called Lambert W function. Much more data is needed to verify empirically which one is closer to the true growth rate. Proof of Smarandache Conjecture In this article we intend to prove that there are no equations of type 1.
Figure 1 The functions 2. Let s be the solution to equation 2. Therefore s, the solution for equation 2. Figure 2 The functions 2. Let s the equation solution 2. Therefore the solution s, of the equation 2. Remark 2. The Theorem 2. Since function db manifests the fastest growth rate we can state that the function gA increases more rapidly then function gb. Proof The theorem can be proven by direct computation, as observed in the graph from Figure 4.
Proof Again, the theorem can be proven by direct calculation as one can observe from the graph in Figure 4. The columns of Table 2 represent the values of the maximal gaps an , cgn , fn , cn , bn and gn , [14, 2, 28, 15]. This valid statements in respect to the inequality 2. Conjecture 2. We solve the following equation, equivalent to 2. In accordance to Theorem 2.
Therefore if we prove that the solutions to equation 2. If we consider the exceptional cases 2. Smarandache Constant We order the solutions to equation 2. References  D. Andrica, Note on a conjecture in prime number theory, Studia Univ. Babes- Bolyai Math. Guy, Unsolved Problems in Number Theory, 2nd ed. Hardy and W. Wright, An Introduction to the Theory of Numbers,5th ed. Landau, Elementary Number Theory, Celsea, Loo, On the primes in the interval [3n, 4n], Int.
Sciences, 6 , no. Nicely, New maximal prime gaps and first occurrences, Mathematics of Computation, 68 , no. Orsted, G. Forchhammer and J. Steenstrup eds. NT], 2 Apr Rivera, Conjecture Westzynthius, Uber die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind, Commentationes Physico-Mathematicae Helingsfors, 5 , Wolf and M. Wolf, First occurrence of a given gap between consecutive primes, Narahari N.
Abstract: A pseudocoloring of G is a coloring of G in which adjacent vertices can receive the same color. Key Words: Coloring, pseudocoloring, neighborhood, domination. Introduction Historically, the coloring terminology comes from the map-coloring problem which involves coloring of the countries in a map in such a way that no two adjacent countries are colored with the same color.
The committee scheduling problem is another problem which can be rephrased as a vertex coloring problem. As such, the concept of graph coloring motivates varieties of graph labelings with an addition of various conditions and has a wide range of applications - channel assignment in wireless communications, traffic phasing, fleet maintenance and task assignment to name a few. More applications of graph coloring can be found in [2,17]. A detailed discussion on graph coloring and some of its variations can be seen in [1,,16,18]. Throughout this paper, we consider a graph G which is simple, finite and undirected.
A vertex k-coloring c of a graph G is said to be a proper k-coloring if vertices of G receive different colors whenever they are adjacent in G. It can be seen that a proper k-coloring of G is simply a vertex partition of V G S into k independent subsets called color classes.
As discussed in , a dominating set S of a graph G V, E is a subset of V such that every vertex in V is either an element of S or is adjacent to an element of S. As introduced in by Harary et al. In , while working on the famous Nordaus - Gaddum inequality , R. Gupta  introduced a new coloring parameter, called the pseudoachromatic number, which generalizes the achromatic number. A pseudo k-coloring of G is a k-coloring in which adjacent vertices may receive same color.
A pseudocomplete k-coloring of a graph G is a pseudo k-coloring such that, for any pair of distinct colors, there is at least one edge whose end vertices are colored with this pair of colors. This parameter was later studied by V. Bhave , E. Sampath Kumar  and V. Yegnanarayanan . Motivated by the above studies, we introduce here a new graph invariant and study some of its properties in this paper.
We use standard notations, the terms not defined here may be found in [4, 12, 14, 15]. Further, a coloring c for which k is maximum is called a maximal neighborhood pseudocoloring of G.
The Figures show a graph G and its various colorings. The above Definition 1. Observation 1. Preliminary Results In this section, we study the neighborhood pseudochromatic number of standard graphs. We also obtain certain bounds on the neighborhood pseudochromatic number of a graph. We end the section with a few characterizations.
We first state the following theorem whose proof is immediate. If possible, suppose that G has neither a C3 or nor a P4 as its subgraph, then G is isomorphic to K1,n. As a consequence of Lemma 2. Proof If a graph G of order n is totally disconnected, then by Theorem 2. Conversely, if G is not totally disconnected, then G has an edge, say, e. Hence the theorem. By Observation 1. This implies that all vertices but two in V receive different colors under c. Now, for each i, 3 6 i 6 n, we have known that c N [vi ] is an injection unless both v1 and v2 are in N [vi ].
Thus each vi is adjacent to both v1 and v2 in G. In this section, we establish a bound on the neighborhood pseudochromatic number of a graph in terms of its domination number. Lemma 4. Let S1 be the set of all pendant vertices of G in S. It is easily seen that S2 is a dominating set. In this case, S2 is the required set. Now, we replace S by S2 and proceed further. By Lemma 4. This ensures that c is a neighborhood pseudocoloring of G. The result obtained for connected graphs can be easily extended to disconnected graphs.
Observation 5. Figure 6. Proof Let G be a non-trivial connected graph. By Theorem 4. Figure 7. Conclusion In this paper, we have obtained the neighborhood pseudochromatic number of some standard graphs. We have established some trivial lower bounds on this number. Improving on these lower bounds remains an interesting open problem. Acknowledgment The authors are indebted to the learned referees for their valuable suggestions and comments. They are thankful to the Principals, Prof. Nanjundaswamy, Dr. Ambedkar Institute of Technology and Prof. Y, University College of Science, Tumkur University, Tumkur for their constant support and encouragement during the preparation of this paper.
References  Akiyama J. Bertram E. Bhave V. Buckley F. Gallian J. Geetha K. Griggs J. Gupta R. Harary F. Theory 8 , Hartsfield G. Havet F. Haynes T. Jensen T. Nordhaus E. Monthly 63 , Pirzada S. Roberts F. Alavi, G. Chartrand, O. Oellermann and A. Schwenk eds. Sampathkumar E. Yegnanarayanan V. In this paper, we introduce the concept of signed efficient dominating function SEDF for directed graphs.
We study the signed domatic number dS D of directed graphs. Further, we find a necessary and sufficient condition for the existence of SEDF in circulant graphs in terms of covering projection. In , J. Dunbar et al. In , Bohdan Zelinka  extended the concept of signed domination in directed graphs. The maximum number of functions in a signed dominating family on G is the signed domatic number of G, denoted by dS G.
The signed domatic number of undirected and simple graphs was introduced by Volkmann and Zelinka . They determined the signed domatic number of complete graphs and complete bipartite graphs. Further, they obtained some bounds for domatic number. They proved the following results. Theorem 1.
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