In 1967 Harary, Hedetniemi, and Prins showed that every graph G admits a complete t-coloring for t with χ(G)≤t≤ψ(G), where χ(G) denotes the chromatic number of ψ(G) achromatic which is maximum r r-coloring. Recently, Edwards Rza̧żewski (2020) this result fails hypergraphs by proving integer k k≥9, there exists k-uniform hypergraph H χ(H)-coloring ψ(H)-coloring, but no some χ(H)<t<ψ(H). They also...

Given a set of nonnegative integers T , and a function S which assigns a set of integers S(v) to each vertex v of a graph G, an S-list T -coloring c of G is a vertexcoloring (with positive integers) of G such that c(v) ∈ S(v) whenever v ∈ V (G) and |c(u)− c(w)| 6∈ T whenever (u,w) ∈ E(G). For a fixed T , the T -choice number T -ch(G) of a graph G is the smallest number k such that G has an S-li...

Let G = (V (G), E(G)) be a multigraph. The degree of a vertex x in G is denoted by d(x), the greatest degree of a vertex – by ∆(G), the chromatic index of G – by χ ′ (G). Let R ⊆ V (G). An interval (respectively, continuous) on R t-coloring of a multigraph G is a proper coloring of edges of G with the colors 1, 2,. .. , t, in which each color is used at least for one edge, and the edges inciden...

Let H s → G mean that every s-coloring of E(H) produces a monochromatic copy of G in some color class. Let the s-color degree Ramsey number of a graph G, written R∆(G; s), be min{∆(H) : H s → G}. If T is a tree in which one vertex has degree at most k and all others have degree at most ⌈k/2⌉, then R∆(T ; s) = s(k− 1) + ǫ, where ǫ = 1 when k is odd and ǫ = 0 when k is even. For general trees, R∆...

An optimal algorithm is presented about Conflict-Free Coloring for connected subgraphs of tree of rings. Suppose the number of the rings in the tree is |T| and the maximum length of rings is |R|. A presented algorithm in [1] for a Tree of rings used O(log|T|.log|R|) colors but this algorithm uses O(log|T|+log|R|) colors. The coloring earned by this algorithm has the unique-min property, that is...

An edge-coloring of a graph G with colors 1, 2, . . . , t is called an interval t-coloring if for each i ∈ {1, 2, . . . , t} there is at least one edge of G colored by i, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In this paper we prove that if a connected graph G with n vertices admits an interval t-coloring, then t ≤ 2n − 3. We also show...

Background. Consider a cellular network consisting of a set of base stations, where the signal from a given base station can be received by clients within a certain distance from the base station. In general, these regions will overlap. For a client, this may lead to interference of the signals. Thus one would like to assign frequencies to the base stations such that for any client within reach...

In this paper, we show that injectivity with respect to the class $mathcal{D}$  of dense monomorphisms of an idempotent and weakly hereditary closure operator of an arbitrary category  well-behaves. Indeed, if $mathcal{M}$ is a subclass of monomorphisms, $mathcal{M}cap mathcal{D}$-injectivity  well-behaves. We also introduce the notion of $(r,t)$-injectivity in the category {bf S-Act}, where $r...

A proper edge t-coloring of a graphG is a coloring of edges of G with colors 1, 2, . . . , t such that all colors are used, and no two adjacent edges receive the same color. The set of colors of edges incident with a vertex x is called a spectrum of x. Any nonempty subset of consecutive integers is called an interval. A proper edge t-coloring of a graph G is interval in the vertex x if the spec...