About Vertex Mappings

In [6] partial graph mappings were formalized in the Mizar system [3]. Such mappings map some vertices and edges of a graph to another while preserving adjacency. While this general approach is appropriate for the general form of (multidi)graphs as introduced in [7], a more specialized version for graphs without parallel edges seems convenient. As such, partial vertex mappings preserving adjacency between the mapped verticed are formalized [email protected] Gutenberg University, Mainz, GermanyIan Anderson. A first course in discrete mathematics. Springer Undergraduate Mathematics Series. Springer, London, 2001. ISBN 1-85233-236-0.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. About graph mappings. Formalized Mathematics, 27(3):261–301, 2019. doi:10.2478/forma-2019-0024.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.27330331

Miscellaneous Graph Preliminaries

This article contains many auxiliary theorems which were missing in the Mizar Mathematical Library [2] to the best of the author’s knowledge. Most of them regard graph theory as formalized in the GLIB series (cf. [8]) and most of them are preliminaries needed in [7] or other forthcoming [email protected] Gutenberg University, Mainz, GermanyGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.John Adrian Bondy and U. S. R. Murty. Graph Theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. ISBN 978-1-84628-969-9.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. About graph mappings. Formalized Mathematics, 27(3):261–301, 2019. doi:10.2478/forma-2019-0024.Sebastian Koch. About graph complements. Formalized Mathematics, 28(1):41–63, 2020. doi:10.2478/forma-2020-0004.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.233

Unification of Graphs and Relations in Mizar

A (di)graph without parallel edges can simply be represented by a binary relation of the vertices and on the other hand, any binary relation can be expressed as such a graph. In this article, this correspondence is formalized in the Mizar system [2], based on the formalization of graphs in [6] and relations in [11], [12]. Notably, a new definition of createGraph will be given, taking only a non empty set V and a binary relation E ⊆ V × V to create a (di)graph without parallel edges, which will provide to be very useful in future [email protected] Gutenberg University, Mainz, GermanyGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Sebastian Koch. Underlying simple graphs. Formalized Mathematics, 27(3):237–259, 2019. doi:10.2478/forma-2019-0023.Gilbert Lee and Piotr Rudnicki. Alternative graph structures. Formalized Mathematics, 13(2):235–252, 2005.Karol Pąk. The friendship theorem. Formalized Mathematics, 20(3):235–237, 2012. doi:10.2478/v10037-012-0028-7.Gunther Schmidt and Thomas Ströhlein. Relations and graphs: discrete mathematics for computer scientists. Springer Science & Business Media, 2012.Yozo Toda. The formalization of simple graphs. Formalized Mathematics, 5(1):137–144, 1996.Robin James Wilson. Introduction to Graph Theory. Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.28217318

On the expected number of perfect matchings in cubic planar graphs

A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with $n$ vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically $c\gamma^n$, where $c>0$ and $\gamma \sim 1.14196$ is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.Comment: 19 pages, 4 figure

Route Planning in Transportation Networks

We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4, previously published by Microsoft Research. This work was mostly done while the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at Microsoft Research Silicon Valle

Trivalent graphs, volume conjectures and character varieties

The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum volume conjecture) are extended to U_q(\fraksl_2) colored quantum invariants of the theta and tetrahedron graph. The \SL(2,\bC) character variety of the fundamental group of the complement of a trivalent graph with $E$ edges in $S^3$ is a Lagrangian subvariety of the Hitchin moduli space over the Riemann surface of genus $g=E/3+1$. For the theta and tetrahedron graph, we conjecture that the configuration of the character variety is locally determined by large color asymptotics of the quantum invariants of the trivalent graph in terms of complex Fenchel-Nielsen coordinates. Moreover, the $q$-holonomic difference equation of the quantum invariants provides the quantization of the character variety.Comment: 11 pages, 2 figure