On fixing elements in matroid minors

LetF be a collection of 3-connected matroids which is (3, 1)-rounded, that is, whenever a 3-connected matroidM has a minor in F ande is an element ofM, thenM has a minor in F whose ground set contains.e. The aim of this note is to prove that, for all sufficiently largen, the collection ofn-element 3-connected matroids having some minor inF is also (3, 1)-rounded.This research was partially supported by the National Science Foundation under Grant No. DMS-8500494.     

Diameter-vital edges in a graph

The graph resulting from contracting edge e is denoted as G/e and the graph resulting from deleting edge e is denoted as G − e. An edge e is diameter-essential if diam(G/e) < diam(G), diameter-increasing if diam(G − e) < diam(G), and diameter-vital if it is both diameter-essential and diameter-increasing. We partition the edges that are not diameter-vital into three categories. In this paper, we study realizability questions relating to the number of edges that are not diameter-vital in the three defined categories. A graph is diameter-vital if all its edges are diameter-vital. We give a structural characterization of diameter-vital graphs.     

Diameter-essential edges in a graph

The eccentricity e(v) of v is the distance to a farthest vertex from v. The diameter diam(G) is the maximum eccentricity among the vertices of G. The contraction of edge e=uv in G consists of removing e and identifying u and v as a single new vertex w, where w is adjacent to any vertex that at least one of u or v were adjacent to. The graph resulting from contracting edge e is denoted G/e. An edge e is diameter-essential if diam(G/e)<diam(G). Let c(G) denote the number of diameter-essential edges in graph G. In this paper, we study existence and extremal problems for c(G); determine bounds on c(G) in terms of diameter and order; and obtain characterizations of graphs achieving extreme values of c(G).     

Eulerian and even-face ribbon graph minors

In this paper, we introduce Eulerian and even-face ribbon graph minors. These minors preserve Eulerian and even-face properties of ribbon graphs, respectively. We then characterize Eulerian, even-face, plane Eulerian and plane even-face ribbon graphs using these minors.     

Excluded checkerboard colourable ribbon graph minors

Motivated by the Eulerian ribbon graph minors, in this paper we introduce the notion of checkerboard colourable minors for ribbon graphs and its dual: bipartite minors for ribbon graphs. Motivated by the bipartite minors of abstract graphs, another bipartite minors for ribbon graphs, i.e. the bipartite ribbon graph join minors are also introduced. Using these minors then we give excluded minor characterizations of the classes of checkerboard colourable ribbon graphs, bipartite ribbon graphs, plane checkerboard colourable ribbon graphs and plane bipartite ribbon graphs.     

On minors of graphs with at least 3n −4 edges

A point disconnecting set S of a graph G is a nontrivial m-separator, where m = |S|, if the connected components of G - S can be partitioned into two subgraphs, each of which has at least two points. A 3-connected graph is quasi 4-connected if it has no nontrivial 3-separators. Suppose G is a graph having n ≥ 6 points. We prove three results: (1) If G is quasi 4-connected with at least 3n - 4 edges, then the graph K^?₁, obtained from K₆ by deleting an edge, is a minor of G. (2) If G has at least 3n - 4 edges then either K^?₆ or the graph obtained by pasting two disjoint copies of K₅ together along a triangle is a minor of G. (3) If the minimum degree of G is at least 6, then K^?₆ is a minor of G. © 1993 John Wiley & Sons, Inc.     

An accessibility theorem for infinite graph minors

We prove the following recent conjecture of Halin. Let Γ₀ be the class of all graphs, and for every ordinal μ > 0 let Γ_μ be the class of all graphs containing infinitely many disjoint connected graphs from Γ_λ, for every λ < μ. Then a graph lies in all these classes Γ_μ if and only if it contains a subdivision of the infinite binary tree. Published by John Wiley & Sons, Inc., 2000 J Graph Theory 35: 273–277, 2000     

Symplectic geometry,minors and graph Laplacians

In my talk, I will present some works done in the nineties on Laplacians on graphs: from eigenvalue problems to inverse problem for resistor networks. I will focus on the motivations and the main results as well as on the main ideas:

• •A differential topology point of view on the minor relation: a nice stratification associated to a finite graph Γ whose strata are associated to the minors of Γ
• •“Discrete” (graphs) versus “continuous” (Riemannian manifolds)
• •Stability of spectra with respect to singular limits: a finite dimensional theory of operators with domains (Von Neumann theory).

The link with topology will appear in some results about my graph parameter μ, in particular the planarity and the linkless embedding properties.     

On the number of edges in the transitive closure of a graph

A property of the square sum of partitions of integers is investigated. The square sum has a direct relation to the number of edges in the transitive closure of a graph. This paper is concerned with the problem of determining the minimum missing value in the sequence of square sums. Asymptotically tight lower and upper bounds on this value are obtained. A consequence of the main result for closure size prediction is also mentioned.     

On 2-factors with cycles containing specified edges in a bipartite graph

Let k≥1 be an integer and G=(V₁,V₂;E) a bipartite graph with |V₁|=|V₂|=n such that n≥2k+2. In this paper it has been proved that if for each pair of nonadjacent vertices x∈V₁ and y∈V₂, , then for any k independent edges e₁,…,e_k of G, G has a 2-factor with k+1 cycles C₁,…,C_k+1 such that e_i∈E(C_i) and |V(C_i)|=4 for each i∈{1,…,k}. We shall also show that the conditions in this paper are sharp.     

Non-contractible edges in a 3-connected graph

An edgee in a 3-connected graphG is contractible if the contraction ofe inG results in a 3-connected graph; otherwisee is non-contractible. In this paper, we prove that the number of non-contractible edges in a 3-connected graph of orderp≥5 is at most $$3p - \left[ {\frac{3}{2}(\sqrt {24p + 25} - 5} \right],$$ and show that this upper bound is the best possible for infinitely many values ofp.     

On the number of vertices and edges of the Buneman graph

In 1971, Peter Buneman proposed a way to construct a tree from a collection of pairwise compatible splits. This construction immediately generalizes to arbitrary collections of splits, and yields a connected median graph, called the Buneman graph. In this paper, we prove that the vertices and the edges of this graph can be described in a very simple way: given a collection of splitsS, the vertices of the Buneman graph correspond precisely to the subsetsS′ ofS such that the splits inS′ are pairwise incompatible and the edges correspond to pairs (S′, S) withS′ as above andS∈S′. Using this characterization, it is much more straightforward to construct the vertices of the Buneman graph than using prior constructions. We also recover as an immediate consequence of this enumeration that the Buneman graph is a tree, that is, that the number of vertices exceeds the number of edges (by one), if and only if any two distinct splits inS are compatible.     

Partitioning the edges of a graph

For any integer m (≥2), it is known that there are simple graphs of maximum valence m whose edges cannot be coloured with m colours in such a way that adjacent edges shall have different colours. We find those values of m and k for which it is true that every simple graph whose maximum valence does not exceed mk can be coloured with m colours in such a way that no colour appears more than k times at any vertex.     

On the number of edges in induced subgraphs of a special distance graph

We obtain new estimates for the number of edges in induced subgraphs of a special distance graph.     

Connectivity,graph minors,and subgraph multiplicity

It is well known that any planar graph contains at most O(n) complete subgraphs. We extend this to an exact characterization: G occurs O(n) times as a subgraph of any planar graph, if and only if G is three-connected. We generalize these results to similarly characterize certain other minor-closed families of graphs; in particular, G occurs O(n) times as a subgraph of the K_b,c-free graphs, b ≥ c and c ≤ 4, iff G is c-connected. Our results use a simple Ramsey-theoretic lemma that may be of independent interest. © 1993 John Wiley & Sons, Inc.     

Coloring the edges of a directed graph

Finding large cliques in a graph is an important problem in applied discrete mathematics. In directed graph a possible corresponding problem is finding large transitive subtournaments. It is well-known that coloring the graph speeds up the clique search in the undirected case. In this paper we propose coloring schemes to speed up the tournament search in the directed case. We prove two complexity results about the proposed colorings. A consequence of these results is that in practical computations we have to be content with approximate greedy coloring algorithms.     

A characterization of some graph classes using excluded minors

In this article we present a structural characterization of graphs without K ₅ and the octahedron as a minor. We introduce semiplanar graphs as arbitrary sums of planar graphs, and give their characterization in terms of excluded minors. Some other excluded minor theorems for 3-connected minors are shown. Communicated by Attila Pethő     

Vertex-disjoint quadrilaterals containing specified edges in a bipartite graph

The theory of vertex-disjoint cycles and 2-factors of graphs is the extension and generation of the well-known Hamiltonian cycles theory and it has important applications in network communication. In this paper we first prove the following result. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n such that n≥2k+1, where k≥1 is an integer. If d(x)+d(y)≥?(4n+2k?1)/3? for each pair of nonadjacent vertices x and y of G with x∈V ₁ and y∈V ₂, then, for any k independent edges e ₁,…,e _k of G, G contains k vertex-disjoint quadrilaterals C ₁,…,C _k such that e _i ∈E(C _i ) for each i∈{1,…,k}. We further show that the degree condition above is sharp. If |V ₁|=|V ₂|=2k, we give degree conditions that G has a 2-factor with k vertex-disjoint quadrilaterals C ₁,…,C _k containing specified edges of G.