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We prove a topological extension of Dirac's theorem suggested by Gowers in 2005: for any connected, closed surface S, we show that any two-dimensional simplicial complex on n vertices in which each pair of vertices belongs to at least n/3 + o(n) facets contains a homeomorph of S spanning all the vertices. This result is asymptotically sharp, and implies in particular that any 3-uniform hypergraph on n vertices with minimum codegree exceeding n/3 + o(n) contains a spanning triangulation of the sphere. [ABSTRACT FROM AUTHOR]

Abstract Any n -vertex 3-graph with minimum codegree at least ⌊ n ∕ 3 ⌋ must have a spanning tight component, but immediately below this threshold it is possible for no tight component to span more than ⌈ 2 n ∕ 3 ⌉ vertices. Motivated by this observation, we ask which codegree forces a tight component of at least any given size. The corresponding function seems to have infinitely many discontinuities, but we provide upper and lower bounds, which asymptotically converge as the function nears the origin. [ABSTRACT FROM AUTHOR]

We construct minor-closed addable families of graphs that are subcritical and contain all planar graphs. This contradicts (one direction of) a well-known conjecture of Noy. [ABSTRACT FROM AUTHOR]

We prove that every vertex transitive, planar, 1-ended, graph covers every graph whose balls of radius r are isomorphic to the ball of radius r in G for a sufficiently large r . It is natural to ask whether this is true for all finitely presented Cayley graphs, but the answer turns out to be negative (de la Salle and Tessera, 2015). [ABSTRACT FROM AUTHOR]

We classify the planar cubic Cayley graphs of connectivity 2, providing an explicit presentation and embedding for each of them. Combined with Georgakopoulos (2017) this yields a complete description of all planar cubic Cayley graphs. [ABSTRACT FROM AUTHOR]

We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of ‘relative compactness’ for such graphs and study sufficient and necessary conditions for this property in terms of various metrics. We then equip graphs satisfying this property with a finite measure and investigate the associated Laplacian and its semigroup. In this context, our results include the trace class property for the semigroup, uniqueness and existence of solutions to the Dirichlet Problem with boundary arising from the natural compactification, an explicit description of the domain of the Dirichlet Laplacian, convergence of the heat semigroup for large times as well as stochastic incompleteness and transience of the corresponding random walk in continuous time. [ABSTRACT FROM AUTHOR]