Related papers: Eulerian 2-Complexes

We consider the problem of computing the Euler characteristic of an abstract simplicial complex given by its vertices and facets. We show that this problem is #P-complete and present two new practical algorithms for computing Euler…

We deal with the symmetries of a (2-term) graded vector space or bundle. Our first theorem shows that they define a (strict) Lie 2-groupoid in a natural way. Our second theorem explores the construction of nerves for Lie 2-categories,…

We make connections of a counting problem of Eulerian cycles for undirected graphs to homological spectral graph theory, and formulate explicitly a trace formula that identifies the number of Eulerian circuits on an Eulerian graph with the…

We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology, such as the fundamental group, coverings and Euler characteristic; Differential…

Based on the Euler-Lagrange cohomology groups $H_{EL}^{(2k-1)}({\cal M}^{2n}) (1 \leqslant k\leqslant n)$ on symplectic manifold $({\cal M}^{2n}, \omega)$, their properties and a kind of classification of vector fields on the manifold, we…

We give a non-abelian analogue of Whitney's 2-isomorphism theorem for graphs. Whitney's theorem states that the cycle space determines a graph up to 2-isomorphism. Instead of considering the cycle space of a graph which is an abelian…

An Eulerian circuit in a directed graph is one of the most fundamental Graph Theory notions. Detecting if a graph $G$ has a unique Eulerian circuit can be done in polynomial time via the BEST theorem by de Bruijn, van Aardenne-Ehrenfest,…

We prove a generalization of the topological Tverberg theorem. One special instance of our general theorem is the following: Let $\Delta$ denote the 8-dimensional simplex viewed as an abstract simplicial complex, and suppose that its…

Let $G$ be a finite simple graph. The line graph $L(G)$ represents the adjacencies between edges of $G$. We define first the line simplicial complex $\Delta_L(G)$ of $G$ containing Gallai and anti-Gallai simplicial complexes…

In this paper, we describe a regular representation given by Cayley theorem for 2-crossed modules of groups and their associated Gray 3-group groupoids with a single 0-cell and equivalently cat2-groups.

We propose an open question that seeks to generalise the Four Colour Theorem from two to three dimensions. As an appetiser, we show that 12 instead of four colours are both sufficient and necessary to colour every 2-complex that embeds in a…

A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…

Following Sullivan's spacial realization of a differential algebra, we construct a universal integrating Lie 2-groupoid for every Lie algebroid. Then We show that unlike Lie algebras which one-to-one correspond to simply connected Lie…

We introduce the notion of a symmetric basis of a vector space equipped with a quadratic form, and provide a sufficient and necessary condition for the existence to such a basis. Symmetric bases are then used to study Cayley graphs of…

Which $2$-regular subgraph $R$ of a cubic graph $G$ can be extended to a cycle double cover of $G$? We provide a condition which ensures that every $R$ satisfying this condition is part of a cycle double cover of $G$. As one consequence, we…

For any positive integer $n$, Lov\'{a}sz-Schrijver, Taniyama and Skopenkov provided examples of simplicial $n$-complexes that inevitably contain a nonsplittable two-component link of $n$-spheres, no matter how they are embedded into the…

Due to Veblen's Theorem, if a connected multigraph $X$ has even degrees at each vertex, then it is Eulerian and its edge set has a partition into cycles. In this paper, we show that an Eulerian multigraph has a unique partition into cycles…

In this paper, we characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems, which generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including flag descent…

It is known that isomorphisms of graph Jacobians induce cyclic bijections on the associated graphs. We characterize when such cyclic bijections can be strengthened to graph isomorphisms, in terms of an easily computed divisor. The result…

A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected $2$-complex every link graph of which is 3-connected…