Connected matchings in special families of graphs.

A connected matching in a graph is a set of disjoint edges such that, for any pair of these edges, there is another edge of the graph incident to both of them. This dissertation investigates two problems related to finding large connected matchings in graphs. The first problem is motivated by a famous and still open conjecture made by Hadwiger stating that every k-chromatic graph contains a minor of the complete graph Kk . If true, Hadwiger\u27s conjecture would imply that every graph G has a minor of the complete graph K n/a(C), where a(G) denotes the independence number of G. For a graph G with a(G) = 2, Thomassé first noted the connection between connected matchings and large complete graph minors: there exists an ? \u3e 0 such that every graph G with a( G) = 2 contains K ?+, as a minor if and only if there exists a positive constant c such that every graph G with a( G) = 2 contains a connected matching of size cn. In Chapter 3 we prove several structural properties of a vertexminimal counterexample to these statements, extending work by Blasiak. We also prove the existence of large connected matchings in graphs with clique size close to the Ramsey bound by proving: for any positive constants band c with c \u3c ¼, there exists a positive integer N such that, if G is a graph with n =: N vertices, 0\u27( G) = 2, and clique size at most bv(n log(n) )then G contains a connected matching of size cn. The second problem concerns computational complexity of finding the size of a maximum connected matching in a graph. This problem has many applications including, when the underlying graph is chordal bipartite, applications to the bipartite margin shop problem. For general graphs, this problem is NP-complete. Cameron has shown the problem is polynomial-time solvable for chordal graphs. Inspired by this and applications to the margin shop problem, in Chapter 4 we focus on the class of chordal bipartite graphs and one of its subclasses, the convex bipartite graphs. We show that a polynomial-time algorithm to find the size of a maximum connected matching in a chordal bipartite graph reduces to finding a polynomial-time algorithm to recognize chordal bipartite graphs that have a perfect connected matching. We also prove that, in chordal bipartite graphs, a connected matching of size k is equivalent to several other statements about the graph and its biadjacency matrix, including for example, the statement that the complement of the latter contains a k x k submatrix that is permutation equivalent to strictly upper triangular matrix

Graph Theory

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The edge version of Hadwiger\u27s conjecture

A well known conjecture of Hadwiger asserts that Kn + 1 is the only minor minimal graph of chromatic number greater than n. In this paper, all minor minimal graphs of chromatic index greater than n are determined. © 2008 Elsevier B.V. All rights reserved

Spin(9)-invariant valuations

The first aim of this thesis is to give a Hadwiger-type theorem for the exceptional Lie group Spin(9). The space of Spin(9)-invariant k-homogeneous valuations is studied through the construction of an exact sequence involving some spaces of differential forms. We present then a description of the spin representation using the properties of the 8-dimensional division algebra of the octonions. Using this description as well as representation-theoretic formulas, we can compute the dimensions of the spaces of differential forms appearing in the exact sequence. Hence we obtain the dimensions of the spaces of k-homogeneous Spin(9)-invariant valuations for k=0,1,...,16. In the second part of this work, we construct one new element for a basis of one of these spaces. It is clear, that the k-th intrinsic volume is also Spin(9)-invariant. The last chapter of this work presents the construction of a new 2-homogeneous Spin(9)-invariant valuation. On a Riemannian manifold (M,g), we construct a valuation by integrating the curvature tensor over the disc bundle. We associate to this valuation on M a family of valuations on the tangent spaces. We show that these valuations are even and homogeneous of degree 2. Moreover, since the valuation on M is invariant under the action of the isometry group of M, the induced valuation on the tangent space in a point p in M is invariant under the action of the stabilisator of p for all p in M. In the special case where M is the octonionic projective plane, this construction yields an even, homogeneous of degree 2, Spin(9)-invariant valuation, whose Klain function is not constant, i.e. which is linearly independent of the second intrinsic volume

On a special case of Hadwiger's conjecture

Hadwiger's Conjecture seems difficult to attack, even in the very special case of graphs G of independence number α (G) = 2. We present some results in this special case

Aspects of distance and domination in graphs.

Thesis (Ph.D.-Mathematics and Applied Mathematics)-University of Natal, 1995.The first half of this thesis deals with an aspect of domination; more specifically, we investigate the vertex integrity of n-distance-domination in a graph, i.e., the extent to which n-distance-domination properties of a graph are preserved by the deletion of vertices, as well as the following: Let G be a connected graph of order p and let oi- S s;:; V(G). An S-n-distance-dominating set in G is a set D s;:; V(G) such that each vertex in S is n-distance-dominated by a vertex in D. The size of a smallest S-n-dominating set in G is denoted by I'n(S, G). If S satisfies I'n(S, G) = I'n(G), then S is called an n-distance-domination-forcing set of G, and the cardinality of a smallest n-distance-domination-forcing set of G is denoted by On(G). We investigate the value of On(G) for various graphs G, and we characterize graphs G for which On(G) achieves its lowest value, namely, I'n(G), and, for n = 1, its highest value, namely, p(G). A corresponding parameter, 1](G), defined by replacing the concept of n-distance-domination of vertices (above) by the concept of the covering of edges is also investigated. For k E {a, 1, ... ,rad(G)}, the set S is said to be a k-radius-forcing set if, for each v E V(G), there exists Vi E S with dG(v, Vi) ~ k. The cardinality of a smallest k-radius-forcing set of G is called the k-radius-forcing number of G and is denoted by Pk(G). We investigate the value of Prad(G) for various classes of graphs G, and we characterize graphs G for which Prad(G) and Pk(G) achieve specified values. We show that the problem of determining Pk(G) is NP-complete, study the sequences (Po(G),Pl(G),P2(G), ... ,Prad(G)(G)), and we investigate the relationship between Prad(G)(G) and Prad(G)(G + e), and between Prad(G)(G + e) and the connectivity of G, for an edge e of the complement of G. Finally, we characterize integral triples representing realizable values of the triples b,i,p), b,l't,i), b,l'c,p), b,l't,p) and b,l't,l'c) for a graph