A note on short cycles in diagraphs

AbstractIn 1977, Caccetta and Haggkvist conjectured that if G is a directed graph with n vertices and minimal outdegree k, then G contains a directed cycle of length at most [n/k]. This conjecture is known to be true for k ⩽ 3. In this paper, we present a proof of the conjecture for the cases k = 4 and k = 5. We also present a new conjecture which implies that of Caccetta and Haggkvist

Spectral and Combinatorial Aspects of Cayley-Crystals

Owing to their interesting spectral properties, the synthetic crystals over lattices other than regular Euclidean lattices, such as hyperbolic and fractal ones, have attracted renewed attention, especially from materials and meta-materials research communities. They can be studied under the umbrella of quantum dynamics over Cayley graphs of finitely generated groups. In this work, we investigate numerical aspects related to the quantum dynamics over such Cayley graphs. Using an algebraic formulation of the "periodic boundary condition" due to Lueck [Geom. Funct. Anal. 4, 455-481 (1994)], we devise a practical and converging numerical method that resolves the true bulk spectrum of the Hamiltonians. Exact results on the matrix elements of the resolvent, derived from the combinatorics of the Cayley graphs, give us the means to validate our algorithms and also to obtain new combinatorial statements. Our results open the systematic research of quantum dynamics over Cayley graphs of a very large family of finitely generated groups, which includes the free and Fuchsian groups.Comment: converging periodic bc for hyperbolic and fractal crystals, tested against exact result

A fusion of charity and commercial investment principles to maximise social investment in South Africa

South Africa faces a raft of social problems, the enormity of which make it impossible for the government to tackle alone. This has necessitated private sector involvement through socially responsible investments (SRI) and charity. Despite the growth of the SRI industry and years of charitable contributions, social investment into the high-impact areas that need it most remains far too low. This study seeks to understand what is holding back social investment, and how to address this. Using grounded theory methodology, the research finds that traditional SRI investors are inappropriate sources of funding and that charitable funds have largely been deployed inefficiently. The proposed solution is for more use to be made of charitable funders, with the disbursement process employing some commercial investment principles in order to facilitate the recycling of capital, resulting in the growth of social investment over time

Reachability in Restricted Chemical Reaction Networks

The popularity of molecular computation has given rise to several models of abstraction, one of the more recent ones being Chemical Reaction Networks (CRNs). These are equivalent to other popular computational models, such as Vector Addition Systems and Petri-Nets, and restricted versions are equivalent to Population Protocols. This paper continues the work on core reachability questions related to Chemical Reaction Networks; given two configurations, can one reach the other according to the system's rules? With no restrictions, reachability was recently shown to be Ackermann-complete, this resolving a decades-old problem. Here, we fully characterize monotone reachability problems based on various restrictions such as the rule size, the number of rules that may create a species (k-source) or consume a species (k-consuming), the volume, and whether the rules have an acyclic production order (feed-forward). We show PSPACE-completeness of reachability with only bimolecular reactions with two-source and two-consuming rules. This proves hardness of reachability in Population Protocols, which was unknown. Further, this shows reachability in CRNs is PSPACE-complete with size-2 rules, which was previously only known with size-5 rules. This is achieved using techniques within the motion planning framework. We provide many important results for feed-forward CRNs where rules are single-source or single-consuming. We show that reachability is solvable in polynomial time if the system does not contain special void or autogenesis rules. We then fully characterize all systems of this type and show that if you allow void/autogenesis rules, or have more than one source and one consuming, the problems become NP-complete. Finally, we show several interesting special cases of CRNs based on these restrictions or slight relaxations and note future significant open questions related to this taxonomy.Comment: This research was supported in part by National Science Foundation Grant CCF-181760

Reachability in Restricted Chemical Reaction Networks

The popularity of molecular computation has given rise to several models of abstraction, one of the more recent ones being Chemical Reaction Networks (CRNs). These are equivalent to other popular computational models, such as Vector Addition Systems and Petri-Nets, and restricted versions are equivalent to Population Protocols. This paper continues the work on core reachability questions related to Chemical Reaction Networks; given two configurations, can one reach the other according to the system\u27s rules? With no restrictions, reachability was recently shown to be Ackermann-complete, this resolving a decades-old problem.Here, we fully characterize monotone reachability problems based on various restrictions such as the rule size, the number of rules that may create a species (k-source) or consume a species (k-consuming), the volume, and whether the rules have an acyclic production order (feed-forward). We show PSPACE-completeness of reachability with only bimolecular reactions with two-source and two-consuming rules. This proves hardness of reachability in Population Protocols, which was unknown. Further, this shows reachability in CRNs is PSPACE-complete with size-2 rules, which was previously only known with size-5 rules. This is achieved using techniques within the motion planning framework.We provide many important results for feed-forward CRNs where rules are single-source or single-consuming. We show that reachability is solvable in polynomial time if the system does not contain special void or autogenesis rules. We then fully characterize all systems of this type and show that if you allow void/autogenesis rules, or have more than one source and one consuming, the problems become NP-complete. Finally, we show several interesting special cases of CRNs based on these restrictions or slight relaxations and note future significant open questions related to this taxonomy