Scrambling index set of primitive digraphs

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.

Coefficients of ergodicity and the scrambling index

Primitive digraphs with the largest scrambling index

Let k be a positive integer and G = (V(G), E(G)) be a finite and connected graph. A rainbow vertex k-coloring of G is a function c: V(G) → {1,2,…, k} such that for every two vertices u and v in V(G) there exists a u-v path whose internal vertices have distinct colors. Such path is called a rainbow vertex path. The rainbow vertex connection number of G, denoted by rvc(G). is the smallest positive integer k so that G has a rainbow vertex k-coloring. The distance between two difference vertices u and v in V(G), denoted by d(u,v), is the length of a shortest u-v path in G. For i ∈ {1,2,…, k], let Ri be the set of vertices with color i and Π - {R 1,R 2,…,Rk } be fjn ordered partition of V(G). The rainbow code of a vertex v of V(G) with respect to Π is defined as the k-tuple , where d(v, Ri ) =min{d(v, y) | y ∈ Ri } for each i ∈ {1,2,…, k}. If every vertex of G has distinct rainbow codes, then c is called a locating rainbow k-coloring of G. The locating rainbow connection number of G. denoted by rvcl(G). is defined as the smallest positive integer k such that G has a locating rainbow k-coloring. In this paper, we provide the sharp upper and lower bounds for locating rainbow connection number of a graph. We also determine the locating rainbow connection number of some well-known classes of graphs.

Exponents of indecomposability

We construct a nonmetric indecomposable homogeneous continuum with the property that each of its proper subcontinua is an arc.In [5] the second author showed that in the metric case, homogeneous indecomposable continua having only arcs as proper subcontinua are solenoids.In [4] the first author defined a continuum that is indecomposable, nonmetrizable, having only arcs for proper subcontinua.Here we show that this continuum is homogeneous.This answers a question that the first author asked at the Warsaw Geometric Topology Conference in 1978.Let us start by redefining the example.Let S = {elt: t is a real number and i s= v7-1} denote the unit circle in the complex plane.If a = etr and b b= ets, then [a, b) denotes the set {c G S : c = elt and r < t < s}, and (a, b] = ([a, b)u{b})\{a).Let P = Sx{0,l} and let the topology on D be defined by pairs of half-open intervals [a, b) X {1} U (a, b] X {0}, where a, b G S. The space D is sometimes called the double arrow space.Let f:D - D be a homeomorphism defined by f(elt,j) = (elt+l,j).The quotient space K obtained from the space D X [0,1] by identification of points (d, 0) with (f(d), 1), for each d G D, is, as shown in [4], an indecomposable continuum.The same result can be obtained using theorems proved in [3].Let qf be the quotient map from V X [0,1] onto K.We will show that every proper subcontinuum of K is an arc.This argument also will give us another proof that K is indecomposable.Let M be a properbe an open set in K\M such that 0 < u < v < 1 and for some point q(p,r) of M, p does not belong to [a,b] X {0,1}.Assume without loss of generality that p G S X {1}.Let m be the smallest positive integer such that fm(p) G [a, b) X {!}-Let n be the smallest positive integer such that f~n(p) G [a, b) X {1}.Let L be the arc 1f({f-n(p)}X[0,u} U (J {{/fc(p)} X [0,1] : k == -n + 1,..., m -1} U {fm(p)} X [v, 1]) in K\U from qf(f~n(p),u) to qf(fm(p),v).Next we show that M is a subarc of L. To accomplish this let z be a point of K\(L U U).There is a closed-open set D in D such that p G D and f~n(D) (J

A bound on the generalized competition index of a primitive matrix using Boolean rank

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.

The Smarandachely adjacent-vertex total chromatic number of graph G is the smallest positive integer k for which G has a proper total k-coloring such that for any pair of adjacent vertices u,v, the set of colors appearing at vertex u and its incident edges is not a subset of the set of colors appearing at vertex v and its incident edges. In this paper, a kind of Generalized 3-regular Ring Graph is defined, and the Smarandachely adjacent-vertex total chromatic numbers is discussed by using construct coloring function. Furthermore, the conjecture of Smarandachely adjacent-vertex total coloring of 3-regular graph is checked.

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.

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

&lt;p&gt;For any positive integer n, the famous F.Smarandache LCM function SL(n) is&lt;br /&gt; de&macr;ned as the smallest positive integer k.&lt;/p&gt;

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

&lt;p&gt;For a positive integer n. the Smarandache function S(n) is defined as the smallest positive&nbsp;integer such that S(n)! is divisible by n.&lt;/p&gt;

Characterization of irreducible Boolean matrices with the largest generalized competition index