4-methylimidazole (4-mei) is a simple nitrogen-containing heterocyclic compound, and recently classified as group 2b by international agency for research on cancer (iarc). caramel coloring is widely applied food coloring in many food products including coffee, carbonated beverages, beer and wine. in present study, ɣ-irradiation was applied to caramel color iii to reduce the level of 4-mei witho...

<cfμ means: for every coloring c of μ × μ by less than cfμ many colors there are A ⊆ μ with otpA = μ + 1 and B ⊆ μ with otpB = μ such that c is constant on A×B. The proof here is re-arranged slightly differently than the proof in the forthcoming [Sh 513] so that no use of other results of Shelah is made, except for Fact 2 below, which comes from pcf theory. In other words, we avoid here using t...

The L(2, 1)-labeling of a digraph G is a function f from the node set of G to the set of all nonnegative integers such that |f(x) − f(y)| ≥ 2 if x and y are at distance 1, and f(x) 6= f(y) if x and y are at distance 2, where the distance from vertex x to vertex y is the length of a shortest dipath from x to y. The minimum of the maximum used label over all L(2, 1)-labelings of G is called λ(G)....

The theory of mixed hypergraph coloring was first introduced by Voloshin in 1993 and has been growing ever since. The proper coloring of a mixed hypergraph H = (X, C,D) is the coloring of the vertex set X so that no D-hyperedge is monochromatic and no C-hyperedge is polychromatic. A mixed hypergraph with hyperedges of type D, C or B is commonly known as a D, C, or B-hypergraph respectively wher...

4-Methylimidazole (4-MEI) is a simple nitrogen-containing heterocyclic compound, and recently classified as group 2B by International Agency for Research on Cancer (IARC). Caramel coloring is widely applied food coloring in many food products including coffee, carbonated beverages, beer and wine. In present study, ɣ-irradiation was applied to Caramel Color III to reduce the level of 4-MEI witho...

a proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. a graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $g$ such that each vertex receives a color from its own list. in this paper, we prov...

The L(h, k)-labeling is an assignment of non negative integer labels to the nodes of a graph such that ’close’ nodes have labels which differ by at least k, and ’very close’ nodes have labels which differ by at least h. The span of an L(h, k)-labeling is the difference between the largest and the smallest assigned label. We study L(h, k)-labelings of cellular, squared and hexagonal grids, seeki...

Suppose λ,x,ζ traverse the ordered sets Λ, X and Z, respectively and consider the functions f(λ,x,ζ) and g(λ,ζ) satisfying the following conditions, (a) f(λ,x,ζ) > 0 and f is TP2 in each pairs of variables when the third variable is held fixed and (b) g(λ,ζ) is TP2. Then the function h(λ,x) =∫Z f(λ,x,ζ)g(λ,ζ)dµ(ζ), defined on Λ×X is TP2 in (λ,x). The aim of this note is to use a new stochast...

‎For a coloring $c$ of a graph $G$‎, ‎the edge-difference coloring sum and edge-sum coloring sum with respect to the coloring $c$ are respectively‎ ‎$sum_c D(G)=sum |c(a)-c(b)|$ and $sum_s S(G)=sum (c(a)+c(b))$‎, ‎where the summations are taken over all edges $abin E(G)$‎. ‎The edge-difference chromatic sum‎, ‎denoted by $sum D(G)$‎, ‎and the edge-sum chromatic sum‎, ‎denoted by $sum S(G)$‎, ‎a...