Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Gröbner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Gröbner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation.     

On multiplicative graphs and the product conjecture

We study the following problem: which graphsG have the property that the class of all graphs not admitting a homomorphism intoG is closed under taking the product (conjunction)? Whether all undirected complete graphs have the property is a longstanding open problem due to S. Hedetniemi. We prove that all odd undirected cycles and all prime-power directed cycles have the property. The former result provides the first non-trivial infinite family of undirected graphs known to have the property, and the latter result verifies a conjecture of Ne?et?il and Pultr These results allow us (in conjunction with earlier results of Ne?et?il and Pultr [17], cf also [7]) to completely characterize all (finite and infinite, directed and undirected) paths and cycles having the property. We also derive the property for a wide class of 3-chromatic graphs studied by Gerards, [5].     

Invariant means and invariant ideals in L∞(G) for a locally compact group G

For a nondiscrete σ-compact locally compact Hausdorff group G, L_∞(G) is a commutative Banach algebra under pointwise multiplication which has many nonzero proper closed invariant ideals; there is at least a continuum of maximal invariant ideals {N_α} such that N_α1 + N_α2 = L_∞(G) whenever α₁ ≠ α₂. It follows from the construction of these ideals that when G is also amenable as a discrete group, then LIM?TLIM contains at least a continuum of mutually singular elements each of which is singular to any element of TLIM. The supports of left-invariant means are in the maximal ideal space of L_∞(G); the structure of these supports leads to the notion of stationary and transitive maximal ideals. To prove that both these types of maximal ideals are dense among all maximal ideals, one shows that the intersection of all nonzero closed invariant ideals is zero. This is the case even though the intersection of any sequence of closed invariant ideals is not zero and the intersection of all the maximal invariant ideals is not zero.     

On the critical ideals of graphs

We introduce some determinantal ideals of the generalized Laplacian matrix associated to a digraph G, that we call critical ideals of G. Critical ideals generalize the critical group and the characteristic polynomials of the adjacency and Laplacian matrices of a digraph. The main results of this article are the determination of some minimal generator sets and the reduced Gröbner basis for the critical ideals of the complete graphs, the cycles and the paths. Also, we establish a bound between the number of trivial critical ideals and the stability and clique numbers of a graph.     

On the Hilbert function of Artinian local complete intersections of codimension three

In singularity theory or algebraic geometry, it is natural to investigate possible Hilbert functions for special algebras A such as local complete intersections or more generally Gorenstein algebras. The sequences that occur as the Hilbert functions of standard graded complete intersections are well understood classically thanks to Macaulay and Stanley. Very little is known in the local case except in codimension two. In this paper we characterise the Hilbert functions of quadratic Artinian complete intersections of codimension three. Interestingly we prove that a Hilbert function is admissible for such a Gorenstein ring if and only if is admissible for such a complete intersection. We provide an effective construction of a local complete intersection for a given Hilbert function. We prove that the symmetric decomposition of such a complete intersection ideal is determined by its Hilbert function.     

Characterizations of Postman Sets

Using results by McKee and Woodall on binary matroids, we prove that the set of postman sets has odd cardinality, generalizing a result by Toida on the cardinality of cycles in Eulerian graphs. We study the relationship between T-joins and blocks of the underlying graph, obtaining a decom- position of postman sets in terms of blocks. We conclude by giving several characterizations of T-joins which are postman sets and commenting on practical issues.     

Circumference, chromatic number and online coloring

Erd?s conjectured that if G is a triangle free graph of chromatic number at least k≥3, then it contains an odd cycle of length at least k ^2?o(1) [13,15]. Nothing better than a linear bound ([3], Problem 5.1.55 in [16]) was so far known. We make progress on this conjecture by showing that G contains an odd cycle of length at least Ω(k log logk). Erd?s’ conjecture is known to hold for graphs with girth at least five. We show that if a graph with girth four is C ₅ free, then Erd?s’ conjecture holds. When the number of vertices is not too large we can prove better bounds on χ. We also give bounds on the chromatic number of graphs with at most r cycles of length 1 mod k, or at most s cycles of length 2 mod k, or no cycles of length 3 mod k. Our techniques essentially consist of using a depth first search tree to decompose the graph into ordered paths, which are then fed to an online coloring algorithm. Using this technique we give simple proofs of some old results, and also obtain several other results. We also obtain a lower bound on the number of colors which an online coloring algorithm needs to use to color triangle free graphs.     

Join of two graphs admits a nowhere-zero 3-flow

Let G be a graph, and λ the smallest integer for which G has a nowherezero λ-flow, i.e., an integer λ for which G admits a nowhere-zero λ-flow, but it does not admit a (λ ? 1)-flow. We denote the minimum flow number of G by Λ(G). In this paper we show that if G and H are two arbitrary graphs and G has no isolated vertex, then Λ(G ∨ H) ? 3 except two cases: (i) One of the graphs G and H is K ₂ and the other is 1-regular. (ii) H = K ₁ and G is a graph with at least one isolated vertex or a component whose every block is an odd cycle. Among other results, we prove that for every two graphs G and H with at least 4 vertices, Λ(G ∨ H) ? 3.     

Clique-critical graphs: Maximum size and recognition

The clique graph of G, K(G), is the intersection graph of the family of cliques (maximal complete sets) of G. Clique-critical graphs were defined as those whose clique graph changes whenever a vertex is removed. We prove that if G has m edges then any clique-critical graph in K^-1(G) has at most 2m vertices, which solves a question posed by Escalante and Toft [On clique-critical graphs, J. Combin. Theory B 17 (1974) 170-182]. The proof is based on a restatement of their characterization of clique-critical graphs. Moreover, the bound is sharp. We also show that the problem of recognizing clique-critical graphs is NP-complete.     

Complete intersections in spherical varieties

Let G be a complex reductive algebraic group. We study complete intersections in a spherical homogeneous space G / H defined by a generic collection of sections from G-invariant linear systems. Whenever nonempty, all such complete intersections are smooth varieties. We compute their arithmetic genus as well as some of their \(h^{p,0}\) numbers. The answers are given in terms of the moment polytopes and Newton–Okounkov polytopes associated to G-invariant linear systems. We also give a necessary and sufficient condition on a collection of linear systems so that the corresponding generic complete intersection is nonempty. This criterion applies to arbitrary quasi-projective varieties (i.e., not necessarily spherical homogeneous spaces). When the spherical homogeneous space under consideration is a complex torus \((\mathbb {C}^*)^n\), our results specialize to well-known results from the Newton polyhedra theory and toric varieties.     

Complete intersection toric ideals of oriented graphs and chorded-theta subgraphs

Let G=(V,E) be a finite, simple graph. We consider for each oriented graph $G_{\mathcal{O}}$ associated to an orientation ${\mathcal{O}}$ of the edges of G, the toric ideal $P_{G_{\mathcal{O}}}$ . In this paper we study those graphs with the property that $P_{G_{\mathcal{O}}}$ is a binomial complete intersection, for all ${\mathcal{O}}$ . These graphs are called $\text{CI}{\mathcal{O}}$ graphs. We prove that these graphs can be constructed recursively as clique-sums of cycles and/or complete graphs. We introduce chorded-theta subgraphs and some of their properties. Also we establish that the $\text{CI}{\mathcal{O}}$ graphs are determined by the property that each chorded-theta has a transversal triangle. Finally we explicitly give the minimal forbidden induced subgraphs that characterize these graphs, these families of forbidden graphs are: prisms, pyramids, thetas and a particular family of wheels that we call θ-partial wheels.     

Sphericity, cubicity, and edge clique covers of graphs

The sphericity sph(G) of a graph G is the minimum dimension d for which G is the intersection graph of a family of congruent spheres in R^d. The edge clique cover number θ(G) is the minimum cardinality of a set of cliques (complete subgraphs) that covers all edges of G. We prove that if G has at least one edge, then sph(G)?θ(G). Our upper bound remains valid for intersection graphs defined by balls in the L_p-norm for 1?p?∞.     

Some properties of edge intersection graphs of single bend paths on a grid

We consider the family of intersection graphs G of paths on a grid, where every vertex v in G corresponds to a single bend path P_v on a grid, and two vertices are adjacent in G if and only if the corresponding paths share an edge on the grid. We first show that these graphs have the Erdös-Hajnal property. Then we present some properties concerning the neighborhood of a vertex in these graphs, and finally we consider some subclasses of chordal graphs for which we give necessary and sufficient conditions to be edge intersection graphs of single bend paths in a grid.     

On the geometry of complete intersection toric varieties

In this paper we give a geometric characterization of the cones of toric varieties that are complete intersections. In particular, we prove that the class of complete intersection cones is the smallest class of cones which is closed under direct sum and contains all simplex cones. Further, we show that the number of the extreme rays of such a cone, which is less than or equal to 2n − 2, is exactly 2n − 2 if and only if the cone is a bipyramidal cone, where n > 1 is the dimension of the cone. Finally, we characterize all toric varieties whose associated cones are complete intersection cones. Received: 4 July 2005     

Long Cycles Generate the Cycle Space of a Graph

Let G be a 2-connected graph in which the degree of every vertex is at least d. We prove that the cycles of length at least d + 1 generate the cycle space of G, unless G ≌ K_d+1 and d is odd. As a corollary, we deduce that the cycles of length at least d + 1 generate the subspace of even cycles in G. We also establish the existence of odd cycles of length at least d + 1 in the case when G is not bipartite.A second result states: if G is 2-connected with chromatic number at least k, then the cycles of length at least k generate the cycle space of G, unless G ≌ K_k and k is even. Similar corollaries follow, among them a stronger version of a theorem of Erdös and Hajnal.     

A characterization of complete matroid base graphs

This study grew from an attempt to give a local analysis of matroid base graphs. A neighborhood-preserving covering of graphs p:G → H is one such that p restricted to every neighborhood in G is an isomorphism. This concept arises naturally when considering graphs with a prescribed set of local properties. A characterization is given of all connected graphs with two local properties: (a) there is a pair of adjacent points, the intersection of whose neighborhoods does not contain three mutually nonadjacent points; (b) the intersection of the neigh-borhoods of points two apart is a 4-cycle. Such graphs have neighborhoods of the form K_n × K_m for fixed n, m and are either complete matroid base graphs or are their images under neighborhood-preserving coverings. If n ≠ m, the graph is unique; if n = m, there are n ? 3 such images which are nontrivial. These examples prove that no set of properties of bounded diameter can characterize matroid base graphs.     

Triangle-free strongly circular-perfect graphs

Zhu [X. Zhu, Circular-perfect graphs, J. Graph Theory 48 (2005) 186-209] introduced circular-perfect graphs as a superclass of the well-known perfect graphs and as an important χ-bound class of graphs with the smallest non-trivial χ-binding function χ(G)≤ω(G)+1. Perfect graphs have been recently characterized as those graphs without odd holes and odd antiholes as induced subgraphs [M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem, Ann. Math. (in press)]; in particular, perfect graphs are closed under complementation [L. Lovász, Normal hypergraphs and the weak perfect graph conjecture, Discrete Math. 2 (1972) 253-267]. To the contrary, circular-perfect graphs are not closed under complementation and the list of forbidden subgraphs is unknown.We study strongly circular-perfect graphs: a circular-perfect graph is strongly circular-perfect if its complement is circular-perfect as well. This subclass entails perfect graphs, odd holes, and odd antiholes. As the main result, we fully characterize the triangle-free strongly circular-perfect graphs, and prove that, for this graph class, both the stable set problem and the recognition problem can be solved in polynomial time.Moreover, we address the characterization of strongly circular-perfect graphs by means of forbidden subgraphs. Results from [A. Pêcher, A. Wagler, On classes of minimal circular-imperfect graphs, Discrete Math. (in press)] suggest that formulating a corresponding conjecture for circular-perfect graphs is difficult; it is even unknown which triangle-free graphs are minimal circular-imperfect. We present the complete list of all triangle-free minimal not strongly circular-perfect graphs.     

A class of h-perfect graphs

A graph is said to be h-perfect if the convex hull of its independent sets is defined by the constraints corresponding to cliques and odd holes, and the nonnegativity constraints. Series-parallel graphs and perfect graphs are h-perfect. The purpose of this paper is to extend the class of graphs known to be h-perfect. Thus, given a graph which is the union of a bipartite graph G₁ and a graph G₂ having exactly two common nodes a and b, and no edge in common, we prove that G is h-perfect if so is the graph obtained from G by replacing G₁ by an a-b chain (the length of which depends on G₁). This result enables us to prove that the graph obtained by substituting bipartite graphs for edges of a series-parallel graph is h-perfect, and also that the identification of two nodes of a bipartite graph yields an h-perfect graph (modulo a reduction which preserves h-perfection).     

Multigraded commutative algebra of graph decompositions

The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe how to obtain generating sets of toric fiber products in non-zero codimension and discuss persistence of normality and primary decompositions under toric fiber products. Several applications are discussed, including (a) the construction of Markov bases of hierarchical models in many new cases, (b) a new proof of the quartic generation of binary graph models associated to K ₄-minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals.