Eulerian Digraph Research Articles

A geometric proof for the root-independence of the greedoid polynomial of Eulerian branching greedoids

We define the root polytope of a regular oriented matroid, and show that the greedoid polynomial of an Eulerian branching greedoid rooted at vertex v0 is equivalent to the h⁎-polynomial of the root polytope of the dual of the graphic matroid.As the definition of the root polytope is independent of the vertex v0, this gives a geometric proof for the root-independence of the greedoid polynomial for Eulerian branching greedoids, a fact which was first proved by Swee Hong Chan, Kévin Perrot and Trung Van Pham using sandpile models. We also obtain that the greedoid polynomial does not change if we reverse every edge of an Eulerian digraph.

Immersion of complete digraphs in Eulerian digraphs

A digraph G immerses a digraph H if there is an injection f: V(H) → V(G) and a collection of pairwise edge-disjoint directed paths Puv, for uv ∈ E(H), such that Puv starts at f(u) and ends at f(v). We prove that every Eulerian digraph with minimum out-degree t immerses a complete digraph on Ω(t) vertices, thus answering a question of DeVos, McDonald, Mohar and Scheide.

Delta invariant for Eulerian digraphs

For a connected Eulerian digraph, we define a delta invariant by its Laplacian matrix. We present characterizations and algorithms for it. We also compute delta invariant for some examples to illustrate its applications.

A character approach to directed genus distribution of graphs: The bipartite single-black-vertex case

Given an Eulerian digraph, we consider the genus distribution of its face-oriented embeddings. We prove that such distribution is log-concave for two families of Eulerian digraphs, thus giving a positive answer for these families to a question asked in Bonnington et al. (2002) [1]. Our proof uses real-rooted polynomials and the representation theory of the symmetric group Sn. The result is also extended to some factorizations of the identity in Sn that are rotation systems of some families of one-face constellations.

Bipartite spanning sub(di)graphs induced by 2‐partitions

AbstractFor a given ‐partition of the vertices of a (di)graph , we study properties of the spanning bipartite subdigraph of induced by those arcs/edges that have one end in each . We determine, for all pairs of nonnegative integers , the complexity of deciding whether has a 2‐partition such that each vertex in (for ) has at least (out‐)neighbours in . We prove that it is ‐complete to decide whether a digraph has a 2‐partition such that each vertex in has an out‐neighbour in and each vertex in has an in‐neighbour in . The problem becomes polynomially solvable if we require to be strongly connected. We give a characterisation of the structure of ‐complete instances in terms of their strong component digraph. When we want higher in‐degree or out‐degree to/from the other set, the problem becomes ‐complete even for strong digraphs. A further result is that it is ‐complete to decide whether a given digraph has a ‐partition such that is strongly connected. This holds even if we require the input to be a highly connected eulerian digraph.

Dyck-Eulerian digraphs

We introduce a family of Eulerian digraphs, E, associated with Dyck words. We provide the algorithms implementing the bijection between E and W, the set of Dyck words. To do so, we exploit a binary matrix, that we call Dyck matrix, representing the cycles of an Eulerian digraph.

Enumerating Unlabelled Embeddings of Digraphs

AbstractA 2-cell embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. In this paper, a method is developed with the purpose of enumerating unlabelled embeddings for an Eulerian digraph. As an application, we obtain explicit formulas for the number of unlabelled embeddings of directed bouquets of cycles Bn, directed dipoles OD2n and for a class of regular tournaments T2n+1.

Eulerian Circuits with No Monochromatic Transitions in Edge-Colored Digraphs with all Vertices of Outdegree Three

A colored eulerian digraph is an eulerian digraph $G$ where a color is assigned to the tail of each edge and a color is assigned to the head of each edge. A compatible circuit is an eulerian circuit such that for every two consecutive edges $uv$ and $vw$ of the circuit, the color of the head of $uv$ is different from the color of the tail of $vw$. Let $S_3$ be the set of vertices of outdegree and indegree three that have exactly three colors on the incident edges where each color appears on exactly one incoming and exactly one outgoing edge. In this paper we consider graphs where all the vertices are in $S_3$. We show that in several special cases we can determine if a graph has a compatible circuit. Our characterization in these cases give rise to a polynomial-time algorithm that determines the existence of a compatible circuit and provides a compatible circuit if one exists.

Chip-Firing Game and a Partial Tutte Polynomial for Eulerian Digraphs

The Chip-firing game is a discrete dynamical system played on a graph, in which chips move along edges according to a simple local rule. Properties of the underlying graph are of course useful to the understanding of the game, but since a conjecture of Biggs that was proved by Merino López, we also know that the study of the Chip-firing game can give insights on the graph. In particular, a strong relation between the partial Tutte polynomial $T_G(1,y)$ and the set of recurrent configurations of a Chip-firing game (with a distinguished sink vertex) has been established for undirected graphs. A direct consequence is that the generating function of the set of recurrent configurations is independent of the choice of the sink for the game, as it characterizes the underlying graph itself. In this paper we prove that this property also holds for Eulerian directed graphs (digraphs), a class on the way from undirected graphs to general digraphs. It turns out from this property that the generating function of the set of recurrent configurations of an Eulerian digraph is a natural and convincing candidate for generalizing the partial Tutte polynomial $T_G(1,y)$ to this class. Our work also gives some promising directions of looking for a generalization of the Tutte polynomial to general digraphs.

Diameter and maximum degree in Eulerian digraphs

In this paper we consider upper bounds on the diameter of Eulerian digraphs containing a vertex of large degree. Define the maximum degree of a digraph to be the maximum of all in-degrees and out-degrees of its vertices. We show that the diameter of an Eulerian digraph of order n and maximum degree Δ is at most n−Δ+3, and this bound is sharp. We also show that the bound can be improved for Eulerian digraphs with no 2-cycle to n−2Δ+O(Δ2/3), and we exhibit an infinite family of such digraphs of diameter n−2Δ+Ω(Δ1/2). We further show that for bipartite Eulerian digraphs with no 2-cycle the diameter is at most n−2Δ+3, and that this bound is sharp.

Distance and size in digraphs

We determine the maximum number of arcs in an Eulerian digraph of given order and diameter. Our bound generalises a classical result on the maximum number of edges of an undirected graph of given order and diameter by Ore (1968) and Homenko and Ostroverhii˘ (1970). We further determine the maximum size of a bipartite digraph of given order and radius.

Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

A maximal minor M of the Laplacian of an n-vertex Eulerian digraph Γ gives rise to a finite group Zn−1/Zn−1M known as the sandpile (or critical) groupS(Γ) of Γ. We determine S(Γ) of the generalized de Bruijn graphsΓ=DB(n,d) with vertices 0,…,n−1 and arcs (i,di+k) for 0≤i≤n−1 and 0≤k≤d−1, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs.Moreover, for a prime p and an n-cycle permutation matrix X∈GLn(p) we show that S(DB(n,p)) is isomorphic to the quotient by 〈X〉 of the centralizer of X in PGLn(p). This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS, and allows one to speculate upon a possibility to construct normal bases in the finite field Fpn from spanning trees in DB(n,p).

Minimum orders of eulerian oriented digraphs with given diameter

A digraph D is oriented if it does not contain 2-cycles. If an oriented digraph D has a directed eulerian path, it is an oriented eulerian digraph. In this paper, when an oriented eulerian digraph D has minimum out-degree 2 and a diameter d, we find the minimum order of D. In addition, when D is 2-regular with diameter 4m (m ≥ 2), we classify the extremal cases.

Enumeration of digraph embeddings

A cellular embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. The directed genus polynomial of an Eulerian digraph D is the polynomial ΓD(x)=∑h≥0gh(D)xh where gh(D) is the number of directed embeddings into the orientable surface Sh, of genus h, for h=0,1,…. The sequence {gh(D)∣h≥0}, which is called the directed genus distribution of the digraph D, is known for very few classes of graphs, compared to the genus distribution of a graph. This paper introduces a variety of methods for calculating the directed genus distributions of Eulerian digraphs. We use them to derive an explicit formula for the directed genus distribution of any 4-regular outerplanar digraph. We show that the directed genus distribution of such a digraph is determined by the red–blue star decompositions of the characteristic tree for an outerplanar embedding. The directed genus distribution of a 4-regular outerplanar digraph is proved to be log-concave, which is consistent with an affirmative answer to a question of Bonnington et al. (2002). Indeed, the corresponding genus polynomial is real-rooted. We introduce Eulerian splitting at a vertex of a digraph, and we prove a splitting theorem for digraph embedding distributions that is analogous to the splitting theorem for (undirected) graph embedding distributions. This new splitting theorem allows conversion of the enumeration of embeddings of a digraph with vertex degrees larger than 4 into a problem of enumerating the embeddings of some 4-regular digraphs.

Large Feedback Arc Sets, High Minimum Degree Subgraphs, and Long Cycles in Eulerian Digraphs

A minimum feedback arc set of a directed graph G is a smallest set of arcs whose removal makes G acyclic. Its cardinality is denoted by β(G). We show that a simple Eulerian digraph with n vertices and m arcs has β(G) ≥ m2/2n2+m/2n, and this bound is optimal for infinitely many m, n. Using this result we prove that a simple Eulerian digraph contains a cycle of length at most 6n2/m, and has an Eulerian subgraph with minimum degree at least m2/24n3. Both estimates are tight up to a constant factor. Finally, motivated by a conjecture of Bollobás and Scott, we also show how to find long cycles in Eulerian digraphs.

Eulerian Circuits with No Monochromatic Transitions in Edge-colored Digraphs

Let $G$ be an Eulerian digraph with a fixed edge coloring (not necessarily a proper edge coloring). A compatible circuit of $G$ is an Eulerian circuit such that every two consecutive edges in the circuit have different colors. We characterize the existence of compatible circuits for directed graphs avoiding certain vertices of outdegree three. Our result is analogous to a result of Kotzig for compatible circuits in edge-colored Eulerian undirected graphs. From our characterization for digraphs we develop a polynomial time algorithm that determines the existence of a compatible circuit in an edge-colored Eulerian digraph and produces a compatible circuit if one exists. Our results use the fact that rainbow spanning trees have been characterized in edge-colored undirected multigraphs. We provide another graph-theoretical proof of this fact.

Eulerian digraphs and toric Calabi-Yau varieties

We investigate the structure of a simple class of affine toric Calabi-Yau varieties that are defined from quiver representations based on finite eulerian directed graphs (digraphs). The vanishing first Chern class of these varieties just follows from the characterisation of eulerian digraphs as being connected with all vertices balanced. Some structure theory is used to show how any eulerian digraph can begenerated by iterating combinations of just a few canonical graph-theoretic moves. We describe the effect of each of these moves on the lattice polytopes which encode the toric Calabi-Yau varieties and illustrate the construction in several examples. We comment on physical applications of the construction in the context of moduli spaces for superconformal gauged linear sigma models.

The directed genus of the de Bruijn graph

Define the directed genus, Γ ( G ) , of an Eulerian digraph G to be the minimum value of p for which G has a 2-cell embedding in the orientable surface of genus p so that every face of the embedding is bounded by a directed circuit in G . The directed genus of the de Bruijn graph D n is shown to be Γ ( D n ) = 2 n − 1 + 1 − 1 2 ( n + 2 ) ∑ d | n + 2 ϕ ( d ) 2 n + 2 d .

Obstructions to directed embeddings of Eulerian digraphs in the plane

A 2-cell embedding of an Eulerian digraph in a closed surface is said to be directed if the boundary of each face is a directed closed walk in G. We prove Kuratowski-type theorems about obstructions to directed embeddings of Eulerian digraphs in the plane.

Disjoint Cycles in Eulerian Digraphs and the Diameter of Interchange Graphs

Let R=(r1, …, rm) and S=(s1, …, sn) be nonnegative integral vectors with ∑ri=∑sj. Let A(R, S) denote the set of all m×n {0, 1}-matrices with row sum vector R and column sum vector S. Suppose A(R, S)≠∅. The interchange graphG(R, S) of A(R, S) was defined by Brualdi in 1980. It is the graph with all matrices in A(R, S) as its vertices and two matrices are adjacent provided they differ by an interchange matrix. Brualdi conjectured that the diameter of G(R, S) cannot exceed mn/4. A digraph G=(V, E) is called Eulerian if, for each vertex u∈V, the outdegree and indegree of u are equal. We first prove that any bipartite Eulerian digraph with vertex partition sizes m, n, and with more than (17−1)mn/4 (≈0.78mn) arcs contains a cycle of length at most 4. As an application of this, we show that the diameter of G(R, S) cannot exceed (3+17)mn/16 (≈0.445mn). The latter result improves a recent upper bound on the diameter of G(R, S) by Qian. Finally, we present some open problems concerning the girth and the maximum number of arc-disjoint cycles in an Eulerian digraph.