Combinatorics (math.CO)

Abstract: Given graphs $F$ and $G$, a perfect $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$ that together cover all the vertices in $G$. The study of the minimum degree threshold forcing a perfect $F$-tiling in a graph $G$ has a long history, culminating in the K\"uhn--Osthus theorem [Combinatorica 2009] which resolves this problem, up to an additive constant, for all graphs $F$. In this paper we initiate the study of the analogous question for edge-ordered graphs. In particular, we characterize for which edge-ordered graphs $F$ this problem is well-defined. We also apply the absorbing method to asymptotically determine the minimum degree threshold for forcing a perfect $P$-tiling in an edge-ordered graph, where $P$ is any fixed monotone path.

Abstract: We calculate the Wiener index of the zero-divisor graph of a finite semisimple ring. We also calculate the Wiener complexity of the zero-divisor graph of a finite simple ring and find an upper bound for the Wiener complexity in the semisimple case.

Abstract: We present problems and results that combine graph-minors and coarse geometry. For example, we ask whether every geodesic metric space (or graph) without a fat $H$ minor is quasi-isometric to a graph with no $H$ minor, for an arbitrary finite graph $H$. We answer this affirmatively for a few small $H$. We also present a metric analogue of Menger's theorem and Konig's ray theorem. We conjecture metric analogues of the Erdos--Posa Theorem and Halin's grid theorem.

Abstract: The unraveled ball of radius $r$ centered at a vertex $v$ in a weighted graph $G$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We present a general bound on the maximum spectral radius of unraveled balls of fixed radius in a weighted graph. The weighted degree of a vertex in a weighted graph is the sum of weights of edges incident to the vertex. A weighted graph is called regular if the weighted degrees of its vertices are the same. Using the result on unraveled balls, we prove a variation of the Alon-Boppana theorem for regular weighted graphs.

Abstract: The Critical Theorem, due to Henry Crapo and Gian-Carlo Rota, has been extended and generalised in many ways. In this paper, we describe properties of the characteristic polynomial of a weighted lattice showing that it has a recursive description. We use this to obtain results on critical exponents of $q$-polymatroids. We prove a Critical Theorem for representable $q$-polymatroids and we provide a lower bound on the critical exponent. We show that certain families of rank-metric codes attain this lower bound.

Abstract: In structural rigidity, one studies frameworks of bars and joints in Euclidean space. Such a framework is an articulated structure consisting of rigid bars, joined together at joints around which the bars may rotate. In this paper, we will describe articulated motions of realisations of incidence geometries that uses the terminology of graph of groups, and describe the motions of such a framework using group theory. Our approach allows to model a variety of situations, such as parallel redrawings, scenes, polytopes, realisations of graphs on surfaces, and even unique colourability of graphs. This approach allows a concise description of various dualities in rigidity theory. We also provide a lower bound on the dimension of the infinitesimal motions of such a framework in the special case when the underlying group is a Lie group.

Abstract: It is a remarkable fact that for many statistics on finite sets of combinatorial objects, the roots of the corresponding generating function are each either a complex root of unity or zero. We call such polynomials \textbf{cyclotomic generating functions} (CGF's). Previous work studied the support and asymptotic distribution of the coefficients of several families of CGF's arising from tableau and forest combinatorics. In this paper, we continue these explorations by studying general CGF's from algebraic, analytic, and asymptotic perspectives. We review some of the many known examples of CGF's; describe their coefficients, moments, cumulants, and characteristic functions; and give a variety of necessary and sufficient conditions for their existence arising from probability, commutative algebra, and invariant theory. We further show that CGF's are ``generically'' asymptotically normal, generalizing a result of Diaconis. We include several open problems concerning CGF's.

Abstract: Much of dynamical algebraic combinatorics focuses on global dynamical systems defined via maps that are compositions of local toggle operators. The second author and Roby studied such maps that result from toggling independent sets of a path graph. We investigate a "toric" analogue of this work by analyzing the dynamics arising from toggling independent sets of a cycle graph. Each orbit in the dynamical system can be encoded via a grid of 0s and 1s; two commuting bijections on the set of 1s in this grid produce torsors for what we call the infinite snake group and the finite ouroboros groups. By studying related covering maps, we deduce precise combinatorial properties of the orbits. Because the snake and ouroboros groups are abelian, they define tilings of cylinders and tori by parallelograms, which we also characterize. Many of the ideas developed here should be adaptable both to other toggle actions in combinatorics and to other cellular automata.