Bears with Hats and Independence Polynomials

Consider the following hat guessing game. A bear sits on each vertex of a graph $G$, and a demon puts on each bear a hat colored by one of $h$ colors. Each bear sees only the hat colors of his neighbors. Based on this information only, each bear has to guess $g$ colors and he guesses correctly if his hat color is included in his guesses. The bears win if at least one bear guesses correctly for any hat arrangement. We introduce a new parameter - fractional hat chromatic number $\hat{\mu}$, arising from the hat guessing game. The parameter $\hat{\mu}$ is related to the hat chromatic number which has been studied before. We present a surprising connection between the hat guessing game and the independence polynomial of graphs. This connection allows us to compute the fractional hat chromatic number of chordal graphs in polynomial time, to bound fractional hat chromatic number by a function of maximum degree of $G$, and to compute the exact value of $\hat{\mu}$ of cliques, paths, and cycles

Robert's theorem and graphs on complete lattices

Automata networks, and in particular Boolean networks, are used to model diverse networks of interacting entities. The interaction graph of an automata network is its most important parameter, as it represents the overall architecture of the network. A continuous amount of work has been devoted to infer dynamical properties of the automata network based on its interaction graph only. Robert's theorem is the seminal result in this area; it states that automata networks with an acyclic interaction graph converge to a unique fixed point. The feedback bound can be viewed as an extension of Robert's theorem; it gives an upper bound on the number of fixed points of an automata network based on the size of a minimum feedback vertex set of its interaction graph. Boolean networks can be viewed as self-mappings on the power set lattice of the set of entities. In this paper, we consider self-mappings on a general complete lattice. We make two conceptual contributions. Firstly, we can view a digraph as a residuated mapping on the power set lattice; as such, we define a graph on a complete lattice as a residuated mapping on that lattice. We extend and generalise some results on digraphs to our setting. Secondly, we introduce a generalised notion of dependency whereby any mapping $\phi$ can depend on any other mapping $\alpha$. In fact, we are able to give four kinds of dependency in this case. We can then vastly expand Robert's theorem to self-mappings on general complete lattices; we similarly generalise the feedback bound. We then obtain stronger results in the case where the lattice is a complete Boolean algebra. We finally show how our results can be applied to prove the convergence of automata networks

Proceedings of Abstracts Engineering and Computer Science Research Conference 2019

© 2019 The Author(s). This is an open-access work distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. For further details please see https://creativecommons.org/licenses/by/4.0/. Note: Keynote: Fluorescence visualisation to evaluate effectiveness of personal protective equipment for infection control is © 2019 Crown copyright and so is licensed under the Open Government Licence v3.0. Under this licence users are permitted to copy, publish, distribute and transmit the Information; adapt the Information; exploit the Information commercially and non-commercially for example, by combining it with other Information, or by including it in your own product or application. Where you do any of the above you must acknowledge the source of the Information in your product or application by including or linking to any attribution statement specified by the Information Provider(s) and, where possible, provide a link to this licence: http://www.nationalarchives.gov.uk/doc/open-government-licence/version/3/This book is the record of abstracts submitted and accepted for presentation at the Inaugural Engineering and Computer Science Research Conference held 17th April 2019 at the University of Hertfordshire, Hatfield, UK. This conference is a local event aiming at bringing together the research students, staff and eminent external guests to celebrate Engineering and Computer Science Research at the University of Hertfordshire. The ECS Research Conference aims to showcase the broad landscape of research taking place in the School of Engineering and Computer Science. The 2019 conference was articulated around three topical cross-disciplinary themes: Make and Preserve the Future; Connect the People and Cities; and Protect and Care