Given a finite configuration of points in a metric space, a Steiner center (respectively, a centroid) is the point of the space (respectively, of the configuration) that minimizes the sum of the distances from all its elements. Working on the k-dimensional real space endowed with the Manhattan distance, we study the approximate algorithm that takes a point of minimum distance from the Steiner c...

Recently a new graph convexity was introduced, arising from Steiner intervals in graphs that are a natural generalization of geodesic intervals. The Steiner tree of a set W on k vertices in a connected graph G is a tree with the smallest number of edges in G that contains all vertices of W . The Steiner interval I(W ) of W consists of all vertices in G that lie on some Steiner tree with respect...

This paper studies a 4-approximation algorithm for k-prize collecting Steiner tree problems. This problem generalizes both k-minimum spanning tree problems and prize collecting Steiner tree problems. Our proposed algorithm employs two 2-approximation algorithms for k-minimum spanning tree problems and prize collecting Steiner tree problems. Also our algorithm framework can be applied to a speci...

Let G = (V,E) be a graph. A k-coloring of a graph G is a labeling f : V (G) → T , where | T |= k and it is proper if the adjacent vertices have different labels. A graph is k-colorable if it has a proper k-coloring. The chromatic number χ(G) is the least k such that G is k-colorable. Here we study chromatic numbers in some kinds of Harary graphs. Mathematics Subject Classification: 05C15

We study the Steiner k-eccentricity on trees, which generalizes previous one in paper [On average 3-eccentricity of arXiv:2005.10319]. achieve much stronger properties for k-ecc tree than that paper. Based this, a linear time algorithm is devised to calculate vertex tree. On other hand, lower and upper bounds index order n are established based novel technique quite different also easier follow...

Let G = (V, E) be a graph with v vertices and e edges. An (a, d)-vertex-antimagic total labeling is a bijection λ from V (G) ∪ E(G) to the set of consecutive integers 1, 2, . . . , v + e, such that the weights of the vertices form an arithmetic progression with the initial term a and common difference d. If λ(V (G)) = {1, 2, . . . , v} then we call the labeling a super (a, d)-vertex-antimagic t...

A vertex-colored graph G is rainbow vertex-connected if two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a vertex-rainbow u−v geodesi...

‎A $mu$-way $(v,k,t)$ $trade$ of volume $m$ consists of $mu$‎ ‎disjoint collections $T_1$‎, ‎$T_2‎, ‎dots T_{mu}$‎, ‎each of $m$‎ ‎blocks‎, ‎such that for every $t$-subset of $v$-set $V$ the number of‎ ‎blocks containing this t-subset is the same in each $T_i (1leq‎ ‎i leq mu)$‎. ‎In other words any pair of collections ${T_i,T_j}$‎, ‎$1leq i< j leq mu‎$ is a $(v,k,t)$ trade of volume $m$. In th...