# Avoiding monochromatic sub-paths in uniform hypergraph paths and cycles.

@article{Billings2020AvoidingMS, title={Avoiding monochromatic sub-paths in uniform hypergraph paths and cycles.}, author={W. Zane Billings and Justin Clifton and Josh Hiller and Tommy Meek and Andrew Penland and Wesley Rogers and Gabriella Smokovich and Andrew Vasquez-Berroteran and Eleni Zamagias}, journal={arXiv: Combinatorics}, year={2020} }

We present a recursive formula for the number of ways to color $j$ vertices blue in an r-uniform hyperpath of size $n$ while avoiding a blue monochromatic sub-hyperpath of length k. We use this result to solve the corresponding problem for $(r-1)$-tight r-uniform paths and loose r-uniform cycles. This generalizes some well known results from reliability engineering and analysis.

## References

SHOWING 1-10 OF 13 REFERENCES

A Recursive Formula for the Reliability of a r-Uniform Complete Hypergraph and Its Applications

- MathematicsMathematical Problems in Engineering
- 2018

The reliability polynomial R(S,p) of a finite graph or hypergraph S=(V,E) gives the probability that the operational edges or hyperedges of S induce a connected spanning subgraph or subhypergraph,…

m-consecutive-k-out-of-n:F systems

- Mathematics
- 1990

An m-consecutive-k-out-of-n:F system, consists of n components ordered on a line; the system fails if and only if there are at least m nonoverlapping runs of k consecutive failed components. Three…

Random hypergraphs and their applications

- Computer Science, MedicinePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2009

A mathematical model of a tripartite structure of users, resources, and tags-labels collaboratively applied by the users to the resources in order to impart meaningful structure on an otherwise undifferentiated database is proposed.

On binomial and circular binomial distributions of orderk forl-overlapping success runs of lengthk

- Mathematics
- 2005

The number ofl-overlapping success runs of lengthk inn trials, which was introduced and studied recently, is presently reconsidered in the Bernoulli case and two exact formulas are derived for its…

Molecular Hypergraphs: The New Representation of Nonclassical Molecular Structures with Polycentric Delocalized Bonds

- Mathematics, Computer ScienceJ. Chem. Inf. Comput. Sci.
- 1995

Graph presentation of molecular structures is widely used in computational chemistry and theoretical chemical research and the lack of a convenient representation for molecules with delocalized polycentric bonds is a substantial drawback.

GERT Analysis of m-Consecutive-k-Out-of-n Systems

- Mathematics, Computer ScienceIEEE Trans. Reliab.
- 2007

This paper presents a graphical evaluation and review technique (GERT) analysis of both types of systems providing closed form explicit formulae for reliability evaluation in a unified manner and it is observed that m-consecutive-at least-k-out-of-n:F systems are more reliable as the number of possible state combinations leading to system's failure are larger in the latter.

Reliability of Consecutive-k-out-of-n:F System

- EngineeringIEEE Transactions on Reliability
- 1981

A reliability diagram with n components in sequence is called consecutive-k-out-of-n:F system if the system fails whenever k consecutive components are failed. This paper presents a recursive formula…

Review of reliability bounds for consecutive-k-out-of-n systems

- Mathematics14th IEEE International Conference on Nanotechnology
- 2014

This paper reviews many lower and upper bounds for consecutive-k-out-of-n systems presented over the last three decades. The reason is a revived interest to accurately estimate the reliability of…

An Introduction To Probability Theory And Its Applications

- Computer Science
- 1950

A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.

Characteristic patterns of cancer incidence: Epidemiological data, biological theories, and multistage models.

- Computer Science, MedicineProgress in biophysics and molecular biology
- 2017

A general modeling framework is presented which is general enough to model most of observed behaviors and is given a model for the effect of aging on the process of carcinogenesis.