Coverings and matchings in r-partite hypergraphs

Ryser\u27s conjecture postulates that for r -partite hypergraphs, τ ≤ (r - 1)ν where τ is the covering number of the hypergraph and ν is the matching number. Although this conjecture has been open since the 1960s, researchers have resolved it for special cases such as for intersecting hypergraphs where r ≤ 5. In this article, we prove several results pertaining to matchings and coverings in r -partite intersecting hypergraphs. First, we prove that finding a minimum cardinality vertex cover for an r -partite intersecting hypergraph is NP-hard. Second, we note Ryser\u27s conjecture for intersecting hypergraphs is easily resolved if a given hypergraph does not contain a particular subhypergraph, which we call a “tornado.” We prove several bounds on the covering number of tornados. Finally, we prove the integrality gap for the standard integer linear programming formulation of the maximum cardinality r -partite hypergraph matching problem is at least r - k where k is the smallest positive integer such that r - k is a prime power

List coloring in the absence of a linear forest.

The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The Listk-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u)⊆{1,…,k}. Let Pn denote the path on n vertices, and G+H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that Listk-Coloring can be solved in polynomial time for graphs with no induced rP1+P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1+P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H

Open problems on graph coloring for special graph classes.

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring

Hypergraph domination and strong independence

We solve several conjectures and open problems from a recent paper by Acharya [2]. Some of our results are relatives of the Nordhaus–Gaddum theorem, concerning the sum of domination parameters in hypergraphs and their complements. (A dominating set in H is a vertex set D ⊆ X such that, for every vertex x ∈ X \D there exists an edge E ∈ E with x ∈ E and E ∩D ̸ = ∅.) As an example, it is shown that the tight bound γγ(H)+γγ(H) ≤ n+2 holds in hypergraphs H = (X, E) of order n ≥ 6, where H is defined as H = (X, E) with E = {X \ E | E ∈ E}, and γγ is the minimum total cardinality of two disjoint dominating sets. We also present some simple constructions of balanced hypergraphs, disproving conjectures of the aforementioned paper concerning strongly independent sets. (Hypergraph H is balanced if every odd cycle in H has an edge containing three vertices of the cycle; and a set S ⊆ X is strongly independent if |S ∩ E | ≤ 1 for all E ∈ E.

A linear programming-based strategy to save pipette tips in automated DNA assembly

Laboratory automation and mathematical optimization are key to improving the efficiency of synthetic biology research. While there are algorithms optimizing the construct designs and synthesis strategies for DNA assembly, the optimization of how DNA assembly reaction mixes are prepared remains largely unexplored. Here, we focus on reducing the pipette tip consumption of a liquid-handling robot as it delivers DNA parts across a multi-well plate where several constructs are being assembled in parallel. We propose a linear programming formulation of this problem based on the capacitated vehicle routing problem, as well as an algorithm which applies a linear programming solver to our formulation, hence providing a strategy to prepare a given set of DNA assembly mixes using fewer pipette tips. The algorithm performed well in randomly generated and real-life scenarios concerning several modular DNA assembly standards, proving capable of reducing the pipette tip consumption by up to 59% in large-scale cases. Combining automatic process optimization and robotic liquid-handling, our strategy promises to greatly improve the efficiency of DNA assembly, either used alone or combined with other algorithmic DNA assembly optimization methods

Equality of domination and transversal numbers in hypergraphs

A subset <i>S</i> of the vertex set of a hypergraph ℋ is called a dominating set of ℋ if for every vertex <i>v</i> not in <i>S</i> there exists <i>u ∈ S</i> such that <i>u</i> and <i>v</i> are contained in an edge in ℋ. The minimum cardinality of a dominating set in ℋ is called the domination number of ℋ and is denoted by γ(ℋ). A transversal of a hypergraph ℋ is defined to be a subset <i>T</i> of the vertex set such that <i>T ⋂ E ≠ Ø</i> for every edge <i>E</i> of ℋ. The transversal number of ℋ, denoted by <i>t</i>.(ℋ), is the minimum number of vertices in a transversal. A hypergraph is of rank <i>k</i> if each of its edges contains at most <i>k</i> vertices. The inequality <i>t</i>(ℋ) = γ(ℋ) is valid for every hypergraph ℋ without isolated vertices. In this paper, we investigate the hypergraphs satisfying <i>t</i>(ℋ) = γ(ℋ), and prove that their recognition problem is NP-hard already on the class of linear hypergraphs of rank 3, while on unrestricted problem instances it lies inside the complexity class ϴ ^p₂. Structurally we focus our attention on hypergraphs in which each subhypergraph ℋ¹ without isolated vertices fulfills the equality <i>t</i>(ℋ¹) = (ℋ¹). We show that if each induced subhypergraph satisfies the equality then it holds for the non-induced ones as well. Moreover, we prove that for every positive integer <i>k</i>, there are only a finite number of forbidden subhypergraphs of rank <i>k</i>, and each of them has domination number at most <i>k</i>

THE MAXIMUM NUMBER OF EDGES IN 2K_2-FREE GRAPHS OF BOUNDED DEGREE

A graph is 2K,-free if it does not contain an independent pair of edges as an induced subgraph. We show that if G is 2K,-free and has maximum degree A(G) = D, then G has at most 5D2/4 edges if D is even. If D is odd, this bound can be improved to (5D *- 20 + 1)/4. The extremal graphs are unique

Sensitive detection of aflatoxin B1 molecules on gold SPR chip surface using functionalized gold nanoparticles

Due to the warm and favourably humid climate of Southern Hungary, the maize is one of the most important crops. The protection against crop damage caused by fusarium and Aspergillus species is essential. Detection of aflatoxin B1 (AFB1) molecules in cereal crops by selective sensors is important, while they can cause serious diseases in humans and animals if they enter the food chain. Our main objective was to develop selective AFB1 sensor with increased sensitivity applying βCD-functionalized gold nanoparticles (AuβCD NPs) in surface plasmon resonance (SPR) measuring apparatus. The nanoparticles ca. 10 nm in diameter were prepared in the presence of thiol-modified cyclodextrin. The adsorption isotherms of AFB1 on bare, thiol-modified cyclodextrin and AuβCD NPs covered Au film surface were calculated using SPR platform. The AFB1 concentration can be quantitatively determined in the 0.001–23.68 ng/mL range. The AuβCD NPs were found to be highly sensitive and exhibited a remarkably low limit of detection (LOD; 1 pg/mL) without using other analytical reagents

Rankings of graphs

A vertex (edge) coloring c : V ! f1; 2; : : : ; tg (c 0 : E ! f1; 2; : : : ; tg) of a graph G = (V; E) is a vertex (edge) t-ranking if for any two vertices (edges) of the same color every path between them contains a vertex (edge) of larger color. The vertex ranking number Ø r (G) (edge ranking number Ø 0 r (G)) is the smallest value of t such that G has a vertex (edge) t-ranking. In this paper we study the algorithmic complexity of the vertex ranking and edge ranking problems. Among others it is shown that Ø r (G) can be computed in polynomial time when restricted to graphs with treewidth at most k for any fixed k. We characterize those graphs where the vertex ranking number Ø r and the chromatic number Ø coincide on all induced subgraphs, show that Ø r (G) = Ø(G) implies Ø(G) = !(G) (largest clique size) and give a formula for Ø 0 r (Kn )

Rankings of graphs

A vertex (edge) coloring c: V!f1; 2;:::;tg (c 0: E!f1; 2;:::; tg) of a graph G =(V;E) isavertex (edge) t-ranking if for any two vertices (edges) of the same color every path between them contains a vertex (edge) of larger color. The vertex ranking number r(G) (edge ranking number 0 r(G)) is the smallest value of t such that G has a vertex (edge) t-ranking. In this paper we study the algorithmic complexityofthevertex ranking and edge ranking problems. Among others it is shown that r(G) can be computed in polynomial time when restricted to graphs with treewidth at most k for any xed k. We characterize those graphs where the vertex ranking number r and the chromatic number coincide on all induced subgraphs, show that r(G) = (G) implies (G) =!(G) (largest clique size) and give a formula for 0 r(K n)