Cyclic Orientations Research Articles

Linear Algebraic Relations among Cardinalities of Sets of Matroid Functions

We introduce a unifying approach for invariants of finite matroids that count mappings to a finite set. The aim of this paper is to show that if the cardinalities of mappings with fixed values on a restricted set satisfy contraction–deletion rules, then there is a relation among them that can be expressed in terms of linear algebra. In this way, we study regular chain groups, nowhere-zero flows and tensions on graphs, and acyclic and totally cyclic orientations of oriented matroids and graphs.

Dualities and reciprocities on graphs on surfaces

We extend the duality between acyclic orientations and totally cyclic orientations on planar graphs to dualities on graphs on orientable surfaces by introducing boundary acyclic orientations and totally bi-walkable orientations. In addition, we provide a reciprocity theorem connecting local tensions and boundary acyclic orientations. Furthermore, we define the balanced flow polynomial which is connected with tension polynomial by duality and with totally bi-walkable orientations by reciprocity.

Topological Properties of Braid-Paths Connected 2-Simplices in Covering Spaces under Cyclic Orientations

In general, the braid structures in a topological space can be classified into algebraic forms and geometric forms. This paper investigates the properties of a braid structure involving 2-simplices and a set of directed braid-paths in view of algebraic as well as geometric topology. The 2-simplices are of the cyclically oriented variety embedded within the disjoint topological covering subspaces where the finite braid-paths are twisted as well as directed. It is shown that the generated homotopic simplicial braids form Abelian groups and the twisted braid-paths successfully admit several varieties of twisted discrete path-homotopy equivalence classes, establishing a set of simplicial fibers. Furthermore, a set of discrete-loop fundamental groups are generated in the covering spaces where the appropriate weight assignments generate multiplicative group structures under a variety of homological formal sums. Interestingly, the resulting smallest non-trivial group is not necessarily unique. The proposed variety of homological formal sum exhibits a loop absorption property if the homotopy path-products are non-commutative. It is considered that the topological covering subspaces are simply connected under embeddings with local homeomorphism maintaining generality.

Asymptotic behavior of acyclic and cyclic orientations of directed lattice graphs

We calculate exponential growth constants describing the asymptotic behavior of several quantities enumerating classes of orientations of arrow variables on the bonds of several types of directed lattice strip graphs G of finite width and arbitrarily great length, in the infinite-length limit, denoted {G}. Specifically, we calculate the exponential growth constants for (i) acyclic orientations, α({G}), (ii) acyclic orientations with a single source vertex, α0({G}), and (iii) totally cyclic orientations, β({G}). We consider several lattices, including square (sq), triangular (tri), and honeycomb (hc). From our calculations, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. To our knowledge, these are the best current bounds on these quantities. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for the exponential growth constants, with fractional uncertainties ranging from O(10−4) to O(10−2). Further, we present exact values of α(tri), α0(tri), and β(hc) and use them to show that our lower and upper bounds on these quantities are very close to these exact values, even for modest strip widths. Results are also given for a nonplanar lattice denoted sqd. We show that α({G}), α0({G}), and β({G}) are monotonically increasing functions of vertex degree for these lattices. A comparison is given of these exponential growth constants with the corresponding exponential growth constant τ({G}) for spanning trees. Our results are in agreement with inequalities following from the Merino–Welsh and Conde–Merino conjectures.

FPT Algorithms to Enumerate and Count Acyclic and Totally Cyclic Orientations

In this paper, we deal with counting and enumerating problems for two types of graph orientations: acyclic and totally cyclic orientations. Counting is known to be #P-hard for both of them. To circumvent this issue, we propose Fixed Parameter Tractable (FPT) algorithms. For the enumeration task, we construct a Binary Decision Diagram (BDD) to represent all orientations of the two kinds, instead of explicitly enumerating them. We prove that the running time of this construction is bounded by O*(2pw2/4+o(pw2)) with respect to the pathwidth pw. We then develop faster FPT algorithms to count acyclic and totally acyclic orientations, running in O*(2bw2/2+o(bw2)) time, where bw denotes the branch-width of the given graph. These counting algorithms are obtained by applying the observations in our enumerating algorithm to branch decomposition.

Study of Exponential Growth Constants of Directed Heteropolygonal Archimedean Lattices

We infer upper and lower bounds on the exponential growth constants $\alpha(\Lambda)$, $\alpha_0(\Lambda)$, and $\beta(\Lambda)$ describing the large-$n$ behavior of, respectively, the number of acyclic orientations, acyclic orientations with a unique source vertex, and totally cyclic orientations of arrows on bonds of several $n$-vertex heteropolygonal Archimedean lattices $\Lambda$. These are, to our knowledge, the best bounds on these growth constants. The inferred upper and lower bounds on the growth constants are quite close to each other, which enables us to derive rather accurate values for the actual exponential growth constants. Combining our new results for heteropolygonal Archimedean lattices with our recent results for homopolygonal Archimedean lattices, we show that the exponential growth constants $\alpha(\Lambda)$, $\alpha_0(\Lambda)$, and $\beta(\Lambda)$ on these lattices are monotonically increasing functions of the lattice coordination number. Comparisons are made with the corresponding growth constants for spanning trees on these lattices. Our findings provide further support for the Merino-Welsh and Conde-Merino conjectures.

Efficient enumeration of graph orientations with sources

An orientation of an undirected graph is obtained by assigning a direction to each of its edges. It is called cyclic when a directed cycle appears, and acyclic otherwise. We study efficient algorithms for enumerating the orientations of an undirected graph. To get the full picture, we consider both the cases of acyclic and cyclic orientations, under some rules specifying which nodes are the sources (i.e. their incident edges are all directed outwards). Our enumeration algorithms use linear space and provide new bounds for the delay, which is the maximum elapsed time between the output of any two consecutively listed solutions. We obtain a delay of O(m) for acyclic orientations and Õ(m) for cyclic ones. When just a single source is specified, these delays become O(m⋅n) and O(m⋅h+h3), respectively, where h is the girth of the graph without the given source. When multiple sources are specified, the delays are the same as in the single source case.

Comparing the Number of Acyclic and Totally Cyclic Orientations with That of Spanning Trees of a Graph

Comparing the Number of Acyclic and Totally Cyclic Orientations with That of Spanning Trees of a Graph

Properties of chromatic polynomials of hypergraphs not held for chromatic polynomials of graphs

In this paper, we present some properties on chromatic polynomials of hypergraphs which do not hold for chromatic polynomials of graphs. We first show that chromatic polynomials of hypergraphs have all integers as their zeros and contain dense real zeros in the set of real numbers. We then prove that for any multigraph G=(V,E), the number of totally cyclic orientations of G is equal to the value of |P(HG,−1)|, where P(HG,λ) is the chromatic polynomial of a hypergraph HG which is constructed from G. Finally we show that the multiplicity of root “0” of P(H,λ) may be at least 2 for some connected hypergraphs H, and the multiplicity of root “1” of P(H,λ) may be 1 for some connected and separable hypergraphs H and may be 2 for some connected and non-separable hypergraphs H.

Cycles in Oriented 3-Graphs

An oriented 3-graph consists of a family of triples (3-sets), each of which is given one of its two possible cyclic orientations. A cycle in an oriented 3-graph is a positive sum of some of the triples that gives weight zero to each 2-set. Our aim in this paper is to consider the following question: how large can the girth of an oriented 3-graph be? We show that there exist oriented 3-graphs whose shortest cycle has length $$\frac{n^2}{2}(1+o(1))$$n22(1+o(1)): this is asymptotically best possible. We also show that there exist 3-tournaments whose shortest cycle has length $$\frac{n^2}{3}(1+o(1))$$n23(1+o(1)), in complete contrast to the case of 2-tournaments.

Nowhere-Zero $${\vec{k}}$$ k → -Flows on Graphs

We introduce and study a multivariate function that counts nowhere-zero flows on a graph G, in which each edge of G has an individual capacity. We prove that the associated counting function is a piecewise-defined polynomial in these capacities, which satisfies a combinatorial reciprocity law that incorporates totally cyclic orientations of G.

Polynomial algorithms for canonical forms of orientations

We introduce canonical forms that represent certain equivalence classes of totally cyclic and acyclic orientations of graphs and present a polynomial algorithms for their constructions. The forms are used in new formulas evaluating tension and flow polynomials on graphs.

The geometry and combinatorics of cographic toric face rings

In this paper we define and study a ring associated to a graph that we call the cographic toric face ring, or simply the cographic ring. The cographic ring is the toric face ring defined by the following equivalent combinatorial structures of a graph: the cographic arrangement of hyperplanes, the Voronoi polytope, and the poset of totally cyclic orientations. We describe the properties of the cographic ring and, in particular, relate the invariants of the ring to the invariants of the corresponding graph. Our study of the cographic ring fits into a body of work on describing rings constructed from graphs. Among the rings that can be constructed from a graph, cographic rings are particularly interesting because they appear in the study of compactified Jacobians of nodal curves.

The Merino–Welsh conjecture holds for series–parallel graphs

The Merino–Welsh conjecture asserts that the number of spanning trees of a graph is no greater than the maximum of the numbers of totally cyclic orientations and acyclic orientations of that graph. We prove this conjecture for the class of series–parallel graphs.

Enumerating colorings, tensions and flows in cell complexes

We study quasipolynomials enumerating proper colorings, nowhere-zero tensions, and nowhere-zero flows in an arbitrary CW-complex X, generalizing the chromatic, tension and flow polynomials of a graph. Our colorings, tensions and flows may be either modular (with values in Z/kZ for some k) or integral (with values in {−k+1,…,k−1}). We obtain deletion–contraction recurrences and closed formulas for the chromatic, tension and flow quasipolynomials, assuming certain unimodularity conditions. We use geometric methods, specifically Ehrhart theory and inside-out polytopes, to obtain reciprocity theorems for all of the aforementioned quasipolynomials, giving combinatorial interpretations of their values at negative integers as well as formulas for the numbers of acyclic and totally cyclic orientations of X.

A note on spanning trees and totally cyclic orientations of 3-connected graphs

A note on spanning trees and totally cyclic orientations of 3-connected graphs

Spanning trees and orientations of graphs

A conjecture of Merino and Welsh says that the number of spanning trees τ (G) of a loopless and bridgeless multigraph G is always less than or equal to either the number a(G) of acyclic orientations, or the number c(G) of totally cyclic orientations, that is, orientations in which every edge is in a directed cycle. We prove that τ (G) ≤ c(G )i fG has at least 4n edges, and that τ (G) ≤ a(G) if G has at most 16n/15 edges. We also prove that τ (G) ≤ a(G) for all multigraphs of maximum degree at most 3 and consequently τ (G) ≤ c(G) for any planar triangulation.

A Bijection for Eulerian-equivalence Classes of Totally Cyclic Orientations

Gioan showed that the number of cycle reversing classes of totally cyclic orientations of a given graph can be calculated as an evaluation of the corresponding Tutte polynomial. We note that the concept of cycle reversing classes of orientations coincides with that of Eulerian-equivalence classes considered by Chen and Stanley, and Kochol. Based on this coincidence, we give a bijective proof of Gioan’s result. Precisely, the main result of the paper is an algorithmic bijection between the set of Eulerian-equivalence classes of totally cyclic orientations and the set of spanning trees without internally active edges.

On cyclically orientable graphs

Graph G is called cyclically orientable (CO) if it admits an orientation in which every simple chordless cycle is cyclically oriented. This family of graphs was introduced by Barot et al. [Cluster algebras of finite type and positive symmetrizable matrices, J. London Math. Soc. (3) 73 (2006) 545–564]. The authors obtained several nice characterizations of CO-graphs, being motivated primarily by their applications in cluster algebras. Here we obtain several new characterizations that provide algorithms for recognizing CO-graphs and obtaining their cyclic orientations in linear time. We show that the edge maximal CO-graphs are 2-trees; that is, G = ( V , E ) is a 2-tree if and only if G is CO and G ′ = ( V , E ′ ) is not CO whenever E is a proper subset of E ′ .

Combinatorial aspects of L-convex polyominoes

We consider the class of L -convex polyominoes, i.e. those polyominoes in which any two cells can be connected with an “ L ” shaped path in one of its four cyclic orientations. The paper proves bijectively that the number f n of L -convex polyominoes with perimeter 2 ( n + 2 ) satisfies the linear recurrence relation f n + 2 = 4 f n + 1 − 2 f n , by first establishing a recurrence of the same form for the cardinality of the “2-compositions” of a natural number n , a simple generalization of the ordinary compositions of n . Then, such 2-compositions are studied and bijectively related to certain words of a regular language over four letters which is in turn bijectively related to L -convex polyominoes. In the last section we give a solution to the open problem of determining the generating function of the area of L -convex polyominoes.