The dichromatic number of D, denoted by χ→(D), is the smallest integer k such that D admits an acyclic k-coloring. We use maderχ→(F) to denote if χ→(D)≥k, then contains a subdivision F. A digraph F called Mader-perfect for every subdigraph F′ F, maderχ→(F′)=|V(F′)|. extend octi digraphs larger class and prove it Mader-perfect, which generalizes result Gishboliner, Steiner Szabó [Dichromatic for...

Nov\'{a}k conjectured in 1974 that for any cyclic Steiner triple systems of order $v$ with $v\equiv 1\pmod{6}$, it is always possible to choose one block from each orbit so the chosen blocks are pairwise disjoint. We consider generalization this conjecture $(v,k,\lambda)$-designs $1 \leq \lambda k-1$. Superimposing multiple copies a symmetric design shows cannot hold all $v$, but we holds whene...

We study the problem of constructing minimum power-p Euclidean k-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most k additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of p (where p ≥ 1), and the cost of ...

In this paper we consider two variations of the minimum cost Steiner problem on a directed acyclic graph G V E with a non negative weight on each edge of E The minimum directed Steiner network problem is de ned as follows Given a set of starting vertices S V and a set of terminating vertices T V nd a subgraph with the minimum total edge weight such that for each starting vertex s there exists a...

W e formulate the problem of multicast tree generation in asymmetric networks as one of computing a directed Steiner tree of minimal cost. We present a new polynomial-time algorithm that provides for tradeoff selection, using a single parameter K , between the tree-cost (Steiner cost) and the runtime efficiency. Using theoretical analysis, we (1 show that it is highly with a performance guarant...

We present a fast algorithm to compute an optimal rec-tilinear Steiner tree for extremal point sets. A point set is extremal if each point lies on the boundary of a rectilinear convex hull of the point set. Our algorithm can be used in homotopic routing in VLSI layout design and it runs in O(k 2 n) time, where n is the size of the point set and k is the size of its rectilinear convex hull.

This note is motivated by recent work Feng et al. (2021) which studies Novák’s conjecture for Steiner Triple Systems and extends it to cyclic 2-designs, more generally 2-designs. Here we consider instead a generalization k -cycle systems: show that in this setting the generalized false ≥ 5 , construct some families of counterexamples arise.

A pentagonal geometry PENT( k , r) is a partial linear space, where every line incident with points, point r lines, and for each x, there precisely those points that are not collinear x. Here we generalize the concept by allowing x to form set of Steiner system S ( 2 w ) whose blocks lines geometry.

The power of Steiner points was studied in a number of different settings in the context of metric embeddings. Perhaps most notably in the context of probabilistic tree embeddings Bartal and Fakcharoenphol et al. [8, 9, 21] used Steiner points to devise near-optimal constructions of such embeddings. However, Konjevod et al. [24] and Gupta [22] demonstrated that Steiner points do not help in thi...

For a block design D, a series of block intersection graphs Gi, or i-BIG(D), i = 0, . . . , k is defined in which the vertices are the blocks of D, with two vertices adjacent if and only if the corresponding blocks intersect in exactly i elements. A silver graph G is defined with respect to a maximum independent set of G, called an α-set. Let G be an r-regular graph and c be a proper (r + 1)-co...