On The Locating Rainbow Connection Number of A Graph

Objectives: To find the rainbow vertex connection number for Line, Central, Middle and Total graphs of Comb Graph. Methods/Statistical Analysis: The methods to find the rainbow vertex connection number of any graph G is quite different from other coloring problems. Findings: The rainbow vertex connection number for line graph and middle graph of comb graph      2 n is rainbow vertex connection number for Central graph of Comb graph is 2 and Total graph of Comb graph is     3 n if n=7, 11; is     3 n -1 if n=4k+11, k ≥ 1. While finding the achromatic number for any graph G , noting that no two adjacent vertices receives same color but in the case of rainbow vertex coloring we can assign same color to the adjacent vertices. Application/Improvements: The applications of rainbow vertex connection number are same as rainbow connection number.

A vertex-colored graph G is rainbow vertex connected if any two distinct vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex connected. In this paper, we prove that for a connected graph G, if \({{\rm diam}(\overline{G}) \geq 3}\), then \({{\rm rvc}(G) \leq 2}\), and this bound is tight. Next, we obtain that for a triangle-free graph \({\overline{G}}\) with \({{\rm diam}(\overline{G}) = 2}\), if G is connected, then \({{\rm rvc}(G) \leq 2}\), and this bound is tight. A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we prove that for a triangle-free graph \({\overline{G}}\) with \({{\rm diam}(\overline{G}) = 3}\), if G is connected, then trc\({(G) \leq 5}\), and this bound is tight. Next, a Nordhaus–Gaddum-type result for the total rainbow connection number is provided. We show that if G and \({\overline{G}}\) are both connected, then \({6 \leq {\rm trc} (G) + {\rm trc}(\overline{G}) \leq 4n - 6.}\) Examples are given to show that the lower bound is tight for \({n \geq 7}\) and n = 5. Tight lower bounds are also given for n = 4, 6.

The rainbow connection was first introduced by Chartrand in 2006 and then in 2009 Krivelevich and Yuster first time introduced the rainbow vertex connection. Let graph be a connected graph. The rainbow vertex-connection is the assignment of color to the vertices of a graph , if every vertex on the graph is connected by a path graph that has interior vertices in different colors. The minimum number of colors from the rainbow vertex coloring in the graph is called rainbow vertex connection number which is denoted . The results of the research are the rainbow vertex connection number of bull graph, net graph, triangular ladder graph, and graph composition (Pn[P1]).

The Ramsey number R(F) of a graph F without isolated vertices is the smallest positive integer n such that every red–blue coloring of Kn produces a subgraph isomorphic to F all of whose edges are colored the same. Let F be a set of graphs without isolated vertices. For a positive integer t, the vertex-disjoint Ramsey number VRt(F) is the smallest positive integer n such that every red–blue coloring of the complete graph Kn of order n results in at least t pairwise vertex-disjoint monochromatic graphs in F; while the edge-disjoint Ramsey number ERt(F) is the smallest positive integer n such that every red–blue coloring of Kn produces at least t pairwise edge-disjoint monochromatic graphs in F. If t=1 and F consists of a single graph F, then VR1(F)=ER1(F)=R(F) is the Ramsey number of the graph F. Thus, the concepts of vertex-disjoint and edge-disjoint Ramsey numbers provide a generalization of the standard Ramsey number. Upper and lower bounds for VRt(F) and ERt(F) are established for sets F of graphs without isolated vertices and the sharpness of these bounds is discussed. The primary goal of this paper is to investigate the values of VRt(F) and ERt(F) for sets F of graphs of size 2 or 3 without isolated vertices. The exact values of VRt(F) are determined for all such sets F and all integers t≥2. The exact values of ERt(F) of certain such sets F with prescribed conditions for all integers t≥2 are determined. For some special sets F of graphs of size 2 or 3 without isolated vertices, the exact values of ERt(F) are determined for 2≤t≤4. Additional results, problems, and conjectures are also presented dealing with these two Ramsey concepts for graphs in general.

Cryptography plays a crucial role in securing information and communications in the face of advancing technologies. Asymmetric encryption, also known as public-key cryptography, plays a crucial role in cryptography. Unlike symmetric encryption, which uses a single key for both encryption and decryption, asymmetric encryption involves a pair of keys, namely a public key and a private key. Asymmetric cryptography is closely associated with the secure management of keys, addresses, and transactions within the blockchain ecosystem, especially in cryptocurrency platform. In this study,we present a novel concept known as rainbow vertex antimagic coloring. This concept extends the idea of rainbow vertex coloring by incorporating antimagic labeling. Let f : E(G) → {1, 2, . . . , |E(G)|} be a function, where the weight of a vertex v ∈ V (G) with respect to f is defined as wf (v) = Σe∈E(v) f(e). Here, E(v) denotes the set of edges incident to v. Thefunction f is termed a vertex antimagic edge labeling if it assigns distinct weights to each vertex. A path is termed a rainbow path if, for any vertices u and v, all internal vertices on the u − v path have distinct weights. The rainbow vertexantimagic connection number of a graph G, denoted by rvac(G), is defined as the minimum number of colors requiredin any rainbow coloring derived from rainbow vertex antimagic labelings of G. In this paper, we will obtain some newlemmas or theorems concerning rvac(G), and we will implement the obtained lemmas or theorems of RVAC on asymmetric cryptography technique.

For a given graph G(V,E), L(1,1,1)-labeling problem is an assignment from vertex set V to the set of non negative integers. If Z^+ be the non negative integers then L(1,1,1)-labeling is a function f:V→Z^+ such that for any two vertices x and y, |f(x)-f(y)|≥1, when d(x,y)=1; |f(x)-f(y)|≥1, when d(x,y)=2; and |f(x)f(y)|≥1, when d(x,y)=3. The L(1,1,1)-chromatic number λ_1,1,1 is the smallest positive integer such that G has an L(1,1,1)-labeling with λ_1,1,1 as the maximum label. In this paper we determine the L(1,1,1)-chromatic number for a path, a cycle, bouquet of cycles joining at a vertex (all are of finite lengths) and sun graph. We also present a lower and upper bounds for λ_1,1,1 in terms of the maximum degree of G.

For a given graph G(V,E), L(1,1,1)-labeling problem is an assignment from vertex set V to the set of non negative integers. If Z^+ be the non negative integers then L(1,1,1)-labeling is a function f:V→Z^+ such that for any two vertices x and y, |f(x)-f(y)|≥1, when d(x,y)=1; |f(x)-f(y)|≥1, when d(x,y)=2; and |f(x)f(y)|≥1, when d(x,y)=3. The L(1,1,1)-chromatic number λ_1,1,1 is the smallest positive integer such that G has an L(1,1,1)-labeling with λ_1,1,1 as the maximum label. In this paper we determine the L(1,1,1)-chromatic number for a path, a cycle, bouquet of cycles joining at a vertex (all are of finite lengths) and sun graph. We also present a lower and upper bounds for λ_1,1,1 in terms of the maximum degree of G.

In discussing unique factorization and ideal theory, C. C. MacDuffee [l, p. 122] cites the multiplicative system of positive integers of the form 1+7& as an example where unique factorization fails, since 792 = 22-36 = 8-99, and 8, 22, 36, 99 are all primes in the system. H. Davenport [2, p. 21] uses positive integers of the form l+ik for the same purpose, with the numerical case 693 = 9-77 = 21-33. In this paper we examine all multiplicative systems made up of arithmetic progressions, and decide the question of unique factorization. For a fixed positive integer ra, let M be a multiplicatively closed system of positive integers such that if xGA2 and y=x (mod ra), y>0, then yG.M. It will be assumed that w is the smallest positive integer which can be used to define M. For example the set M of all positive integers congruent to 1, 3, or 5 modulo 6 is also the set congruent to 1 modulo 2, and in this case ra = 2. We divide the integers 1, 2, • • • , w into two classes: the set A, 0(ra) in number, of those relatively prime to ra, and the others in set B, ra—0(ra) in number.

For any positive integer n, the F.Smarandache LCM function SL(n) is deflned as the smallest positive integer k such that n j (1;2;¢¢¢ ;k), where (1;2;¢¢¢ ;k) denotes the least common multiple of 1;2;¢¢¢ ;k, and let n = p fi 1 1 p fi2 2 ¢¢¢p fis s be the factorization of n into prime powers, then ›(n) = fi1p1 + fi2p2 + ¢¢¢ + fisps. The main purpose of this paper is using the elementary methods to study the mean value properties of ›(n)SL(n), and give a sharper asymptotic formula for it. Keywords F.Smarandache LCM function, ›(n) function, mean value, asymptotic formula.

Let F be a field of characteristic ~ 2 which is not formally real, i.e., 1 is a sum of squares in E Let s be the level ofF,, i.e. s is the smallest positive integer such that 1 is a sum of s squares in F. We shall denote by q the number of square classes of F, i. e. the order of the group F*/(F*) z where F* = F "-. {0}. It is well-known now [2] that s is a power of 2, say s = 2 ~. On the other hand it is clear that q is either infinite or a power of 2. It was shown by Pfister [3] that q>=2 t~(k+l)/2. (1)

<p>The main purpose of this paper is to use the elementary methods to study the mean value of the F.Smarandache LCM function SL(n), and give a sharper asymptotic formula for it.</p>

<p>For any positive integer n, the famous F.Smarandache LCM function SL(n) is<br /> de¯ned as the smallest positive integer k.</p>

<p>For any positive integer n and prime p, let Sp(n) denotes the smallest positive<br /> integer m such that m! is divisible by pn.</p>

<p>For a positive integer n. the Smarandache function S(n) is defined as the smallest positive integer such that S(n)! is divisible by n.</p>

Characterization of irreducible Boolean matrices with the largest generalized competition index