Infinite graphs and bicircular matroids (original) (raw)

Journal of Combinatorial Theory, Series B, 1979

A matroidal family V is defined to be a collection of graphs such that, for any given graph G, the subgraphs of G isomorphic to a graph in V satisfy the matroid circuit-axioms. Here matroidal families closed under homeomorphism are considered. A theorem of Simks-Pereira shows that when only finite connected graphs are allowed as members of Q, two matroids arise: the cycle matroid and bicircular matroid. Here this theorem is generalized in two directions: the graphs are allowed to be infinite, and they are allowed to be disconnected. In the first case four structures result and in the second case two infinite families of rnatroids are obtained. The main theorem concerns the structures resulting when both restrictions are relaxed simultaneously. We will use standard graph theory terminology as far as possible, as found in [l], [2], or [13]. All graphs will be undirected and possibly infinite, and loops and multiple edges will be allowed. If G is a graph, E(G) denotes the set of edges of G and G\e denotes the graph obtained from G by deleting the edge e. A graph H is homeomorphic from G if it is isomorphic to a graph obtained from G by replacing each edge by a finite path and a graph K is homeomorphic to G if there exists some graph H such that G and K are both homeomorphic from H. The matroid theory terminology will follow [12]. One of the many ways to define a matroid on a finite set is by means of its collection % of circuits, which satisfies the following two axioms: (Cl) No member of V properly contains another.

Two matroidal families on the edge set of a graph

Discrete Mathematics, 2002

Let G be a 2-connected undirected graph with n vertices. Its connected subgraphs of n − 1 edges (that is, its spanning trees) are the bases of the usual cycle matroid of G. Let now X be a subset of vertices of G and consider those connected subgraphs of n edges whose unique circuit passes through at least one element of X. They are shown to be the bases of another matroid. A similar construction is given if the connectivity of the subgraph is not required but every circuit of the subgraph must pass through at least one element of X. Both constructions still lead to matroids if X is a subset of edges of G. Relation of the ÿrst construction to elementary strong maps (if G is planar) and representability properties of the matroids arising from these constructions are also presented. Finally, a civil engineering problem is described which served as the original motivation of this study.

Matroids in terms of Cayley graphs and some related results1

The point we try to get across is that the generalization of the coun- terparts of the matroid theory in Cayley graphs since the matroid theory frequently simplify the graphs and so Cayley graphs. We will show that, for a Cayley graph G, the cutset matroid M ( G) is the dual of the circuit matroid M( G). We will also deduce that if G is an abstract-dual of a Cayley graph , then M( G) is isomorphic to (M( G)) . 2000 Mathematics Subject Classication : 05C20; 05C25; 05C60.

Some heterochromatic theorems for matroids

arXiv (Cornell University), 2017

The anti-Ramsey number of Erdös, Simonovits and Sós from 1973 has become a classic invariant in Graph Theory. To study this invariant in Matroid Theory, we use a related invariant introduce by Arocha, Bracho and Neumann-Lara. The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a totally multicoloured hyperedge of H. Given a rank-r matroid M , there are several hypergraphs associated to the matroid that we can consider. One is C(M), the hypergraph where the points are the elements of the matroid and the hyperedges are the circuits of M. The other one is B(M), where here the points are the elements and the hyperedges are the bases of the matroid. We prove that hc(C(M)) equals r + 1 when M is not the free matroid U n,n , and that if M is a paving matroid, then hc(B(M)) equals r. Then we explore the case when the hypergraph has the Hamiltonian circuits of the matroid as hyperedges, if any, for a class of paving matroids. We also extend the trivial observation of Erdös, Simonovits and Sós for the anti-Ramsey number for 3-cycles to 3circuits in projective geometries over finite fields.

On Extremal Connectivity Properties of Unavoidable Matroids

Journal of Combinatorial Theory, Series B, 1999

Ding, Oporowski, Oxley, and Vertigan proved that, for all n 3, there is an integer N(n) such that every 3-connected matroid with at least N(n) elements has a minor isomorphic to a wheel or whirl of rank n, M(K 3, n ) or its dual, U 2, n+2 or its dual, or a rank-n spike. This paper characterizes each of these classes of unavoidable matroids in terms of an extremal connectivity condition. In particular, it is proved that if M is a 3-connected matroid of at least rank 7 for which every single-element deletion or contraction is 3-connected but no 2-element deletion or contraction is, then M is a spike with its tip deleted. It is further proved that if M is a 3-connected matroid of at least rank 4 for which every single-element deletion is 3-connected but no 1-element contraction or 2-element deletion is, then M$M*(K 3, n ).