On the existence of uncountably many matroidal families

A matroidal family is a set $\text{F}$ ≠ ? of connected finite graphs such that for every finite graph G the edge-sets of those subgraphs of G which are isomorphic to some element of $\text{F}$ are the circuits of a matroid on the edge-set of G. Simões-Pereira [5] shows the existence of four matroidal families and Andreae [1] shows the existence of a countably infinite series of matroidal families. In this paper we show that there exist uncountably many matroidal families. This is done by using an extension of Andreae's theorem, a construction theorem, and certain properties of regular graphs. Moreover we observe that all matroidal families so far known can be obtained in a unified way.     

Construction of matroidal families by partly closed sets

A matroidal family of graphs is a set $\text{M}$≠Ø of connected finite graphs such that for every finite graph G the edge sets of those subgraphs of G which are isomorphic to some element of $\text{M}$ are the circuits of a matroid on the edge set of G. In [9], Schmidt shows that, for n?0, ?2n<r?1, n, $\text{r∈}\text{Z}$, the set $\text{M}\text{(n, r)={G∣G}$ is a graph with β(G)=nα(G)+r and α(G )>, and is minimal with this property (with respect to the relation ?))} is a matroidal family of graphs. He also describes a method to construct new matroidal families of graphs by means of so-called partly closed sets. In this paper, an extension of this construction is given. By means of s-partly closed subsets of $\text{M}\text{(n, r)}$, s?r, we are able to give sufficient and necessary conditions for a subset $\text{P}\text{(n, r)}$ of $\text{M}\text{(n, r)}$ to yield a matroidal family of graphs when joined with the set $\text{I}\text{(n, s)}$ of all graphs $\text{G∈}\text{M}\text{(n, s)}$ which satisfy: If $\text{H∈}\text{P}\text{(n, r)}$, then H?G. In particular, it is shown that $\text{M}\text{(n, r)}$ is not a matroidal family of graphs for r?2. Furthermore, for n?0, 1?2n<r, n, $\text{r∈}\text{Z}$, the set of bipartite elements of $\text{M}\text{(n, r)}$ can be used to construct new matroidal families of graphs if and only if s?min(n+r, 1).     

Subgraphs as circuits and bases of matroids

As is well known, the cycles of any given graph G may be regarded as the circuits of a matroid defined on the edge set of G. The question of whether other families of connected graphs exist such that, given any graph G, the subgraphs of G isomorphic to some member of the family may be regarded as the circuits of a matroid defined on the edge set of G led us, in two other papers, to the proof of some results concerning properties of the cycles when regarded as circuits of such matroids. Here we prove that the wheels share many of these properties with the cycles. Moreover, properties of subgraphs which may be regarded as bases of such matroids are also investigated.     

Duality of infinite graphs

Some basic results on duality of infinite graphs are established and it is proven that a block has a dual graph if and only if it is planar and any two vertices are separated by a finite edge cut. Also, the graphs having predual graphs are characterized completely and it is shown that if $\text{G}{}^1$ is a dual and predual graph of G, then G and $\text{G}{}^1$ can be represented as geometric dual graphs. The uniqueness of dual graphs is investigated, in particular, Whitney's 2-isomorphism theorem is extended to infinite graphs. Finally, infinite minimal cuts in dual graphs are studied and the characterization (in terms of planarity and separation properties) of the graphs having dual graphs satisfying conditions on the infinite cuts, as well, is included.     

Matroidal families of finite connected nonhomeomorphic graphs exist

A matroidal family is a nonempty set ? of connected finite graphs such that for every arbitrary finite graph G the edge sets of the subgraphs of G which are isomorphic to an element of ? form a matroid on the edge set of G. In the present paper the question whether there are any matroidal families besides the four previously described by Simões-Pereira is answered affirmatively. It is shown that for every natural number n ? 2 there is a matroidal family that contains the complete graph with n vertices. For n = 4 this settles Simões-Pereira's conjecture that there is a matroidal family containing all wheels.     

On the intersection of infinite matroids

We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved by Aharoni and Berger in 2009.We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids.In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays.     

Connectivity in bicircular matroids

Tutte has defined n-connection for matroids and proved a connected graph is n-connected if and only if its polygon matroid is n-connected. In this paper we introduce a new notion of connection in graphs, called n-biconnection, and prove an analogous theorem for graphs and their bicircular matroids. Results concerning 3-biconnected graphs are also presented.     

A Decomposition Theorem for Binary Matroids with no Prism Minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, K ₅\ e. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac’s infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of F ₇ with itself across a triangle with an element of the triangle deleted; it’s rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3-separation in P ₉. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, F ₇ and PG(3, 2), respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of R ₁₀, the unique splitter for regular matroids. As a corollary, we obtain Mayhew and Royle’s result identifying the binary internally 4-connected matroids with no prism minor Mayhew and Royle (Siam J Discrete Math 26:755–767, 2012).     

Minimal non-orientable matroids of rank three

Minimal non-orientable matroids have been investigated as a threshold between orientability and non-orientability. The Fano matroid and the MacLane matroid are well-known minimal non-orientable matroids of rank 3. A natural question is whether there exists a minimal non-orientable matroid of every rank $r$ with $m$ elements. In this paper, we give an answer to the question in rank 3 that for every $m\ge 7$, there exists a minimal non-orientable matroid of rank 3 with $m$ elements. To prove this statement, we construct two new infinite families of minimal non-orientable matroids of rank 3. Furthermore we also investigate representability of the two infinite families.     

Generalizations of magic graphs

A graph is magic if the edges are labeled with distinct nonnegative real numbers such that the sum of the labels incident to each vertex is the same. Given a graph finite G, an Abelian group $\text{g}$, and an element r(v) ∈ $\text{g}$ for every v ∈ V(G), necessary and sufficient conditions are given for the existence of edge labels from $\text{g}$ such that the sum of the labels incident to v is r(v). When there do exist labels, all possible labels are determined. The matroid structure of the labels is investigated when $\text{g}$ is an integral domain, and a dimensional structure results. Characterizations of several classes of graphs are given, namely, zero magic, semi-magic, and trivial magic graphs.     

Characterization of self-complementary graphs with 2-factors

Let G be a self-complementary graph (s.c.) and π its degree sequence. Then G has a 2-factor if and only if π - 2 is graphic. This is achieved by obtaining a structure theorem regarding s.c. graphs without a 2-factor. Another interesting corollary of the structure theorem is that if G is a s.c. graph of order p?8 with minimum degree at least $\text{p}\text{4}$, then G has a 2-factor and the result is the best possible.     

Independence in graphs with maximum degree four

Let $\text{C}$ be the class of triangle-free graphs with maximum degree four. A lower bound for the number of edges in a graph of $\text{C}$ is derived in terms of its order p and independence β. Also a characterization of certain minimum independence graphs in $\text{C}$ is provided. Let r(k) be the smallest integer such that every graph in $\text{C}$ with at least r(k) vertices has independence at least k. The values of r(k) for all k may be derived from our main theorem and $\text{4}\text{13}$ obtained as the best possible lower bound for the independence ratio $\text{β}\text{p}$ of graphs in $\text{C}$.     

Stability in CAN-free graphs

A class $\text{F}$ of graphs characterized by three forbidden subgraphs C, A, N is considered; C is the claw (the unique graph with degree sequence (1, 1, 1, 3)), A is the antenna (a graph with degree sequence (1, 2, 2, 3, 3, 3) which does not contain C), and N is the net (the unique graph with degree sequence (1, 1, 1, 3, 3, 3)). These graphs are called CAN-free. A construction is described which associates with every CAN-free graph G another CAN-free graph G′ with strictly fewer nodes than G and with stbility number α(G′) = α(G) ? 1. This gives a good algorithm for determining the stability number of CAN-free graphs.     

On a general class of graph polynomials

Let $\text{F}$ be a family of connected graphs. With each element α ∈ $\text{F}$, we can associate a weight w_α. Let G be a graph. An F-cover of G is a spanning subgraph of G in which every component belongs to $\text{F}$. With every F-cover we can associate a monomial π(C) = Π_αw_α, where the product is taken over all components of the cover. The F-polynomial of G is Σπ(C), where the sum is taken over all F-covers in G. We obtain general results for the complete graph and complete bipartite graphs, and we show that many of the well-known graph polynomials are special cases of more general F-polynomials.     

Tough graphs and hamiltonian circuits

The toughness of a graph G is defined as the largest real number t such that deletion of any s points from G results in a graph which is either connected or else has at most s/t components. Clearly, every hamiltonian graph is 1-tough. Conversely, we conjecture that for some t₀, every t₀-tough graph is hamiltonian. Since a square of a k-connected graph is always k-tough, a proof of this conjecture with t₀ = 2 would imply Fleischner's theorem (the square of a block is hamiltonian). We construct an infinite family of ($\text{3}\text{2}$)-tough nonhamiltonian graphs.     

On triangle-free 3-connected matroids

We present a method to construct any triangle-free 3-connected matroid starting from a matroid belonging to one of four infinite families and subsequently performing a sequence of small operations on it. This result extends to matroids a theorem proved by Kriesell for graphs.     

Graphs which,with their complements,have certain clique covering numbers

Two common invariants of a graph G are its node clique cover number, θ₀(G), and its edge clique cover number, θ₁(G). We present in this work a characterization of those graphs for which they and their complements, $\text{G}\text{?}$, have θ₀(G)=θ₁(G) and θ₀($\text{G}\text{?}$)=θ₁($\text{G}\text{?}$). Graphs satis ying these conditions are shown to constitute a subset of those graphs which we term C-graphs.     

Induction of M-convex functions by linking systems

Induction (or transformation) by bipartite graphs is one of the most important operations on matroids, and it is well known that the induction of a matroid by a bipartite graph is again a matroid. As an abstract form of this fact, the induction of a matroid by a linking system is known to be a matroid.M-convex functions are quantitative extensions of matroidal structures, and they are known as discrete convex functions. As with matroids, it is known that the induction of an M-convex function by networks generates an M-convex function. As an abstract form of this fact, this paper shows that the induction of an M-convex function by linking systems generates an M-convex function. Furthermore, we show that this result also holds for M-convex functions on constant-parity jump systems. Previously known operations such as aggregation, splitting, and induction by networks can be understood as special cases of this construction.