A special case of Hadwiger's conjecture

AbstractWe investigate Hadwiger's conjecture for graphs with no stable set of size 3. Such a graph on at least 2t−1 vertices is not t−1 colorable, so is conjectured to have a Kt minor. There is a strengthening of Hadwiger's conjecture in this case, which states that there is a Kt minor in which the preimage of each vertex of Kt is a single vertex or an edge. We prove this strengthened version for graphs with an even number of vertices and fractional clique covering number less than 3. We investigate several possible generalizations and obtain counterexamples for some and improved results from others. We also show that for sufficiently large n, a graph on n vertices with no stable set of size 3 has a K19n4/5 minor using only vertices and single edges as preimages of vertices

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

Complete directed minors and chromatic number

The dichromatic number χ→(D) of a digraph D is the smallest k for which it admits a k-coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of authors have recently studied the containment of complete directed minors in digraphs with a given dichromatic number. In this note we exhibit a relation of these problems to Hadwiger's conjecture. Exploiting this relation, we show that every directed graph excluding the complete digraph K↔t of order t as a strong minor or as a butterfly minor is O(t(log log t)6)-colorable. This answers a question by Axenovich, Girão, Snyder, and Weber, who proved an upper bound of t4t for the same problem. A further consequence of our results is that every digraph of dichromatic number 22n contains a subdivision of every n-vertex subcubic digraph, which makes progress on a set of problems raised by Aboulker, Cohen, Havet, Lochet, Moura, and Thomassé

Balanced-chromatic number and Hadwiger-like conjectures

Motivated by different characterizations of planar graphs and the 4-Color Theorem, several structural results concerning graphs of high chromatic number have been obtained. Toward strengthening some of these results, we consider the \emph{balanced chromatic number}, $\chi_b(\hat{G})$, of a signed graph $\hat{G}$. This is the minimum number of parts into which the vertices of a signed graph can be partitioned so that none of the parts induces a negative cycle. This extends the notion of the chromatic number of a graph since $\chi(G)=\chi_b(\tilde{G})$, where $\tilde{G}$ denotes the signed graph obtained from~$G$ by replacing each edge with a pair of (parallel) positive and negative edges. We introduce a signed version of Hadwiger's conjecture as follows. Conjecture: If a signed graph $\hat{G}$ has no negative loop and no $\tilde{K_t}$-minor, then its balanced chromatic number is at most $t-1$. We prove that this conjecture is, in fact, equivalent to Hadwiger's conjecture and show its relation to the Odd Hadwiger Conjecture. Motivated by these results, we also consider the relation between subdivisions and balanced chromatic number. We prove that if $(G, \sigma)$ has no negative loop and no $\tilde{K_t}$-subdivision, then it admits a balanced $\frac{79}{2}t^2$-coloring. This qualitatively generalizes a result of Kawarabayashi (2013) on totally odd subdivisions

Planar graphs : a historical perspective.

The field of graph theory has been indubitably influenced by the study of planar graphs. This thesis, consisting of five chapters, is a historical account of the origins and development of concepts pertaining to planar graphs and their applications. The first chapter serves as an introduction to the history of graph theory, including early studies of graph theory tools such as paths, circuits, and trees. The second chapter pertains to the relationship between polyhedra and planar graphs, specifically the result of Euler concerning the number of vertices, edges, and faces of a polyhedron. Counterexamples and generalizations of Euler\u27s formula are also discussed. Chapter III describes the background in recreational mathematics of the graphs of K5 and K3,3 and their importance to the first characterization of planar graphs by Kuratowski. Further characterizations of planar graphs by Whitney, Wagner, and MacLane are also addressed. The focus of Chapter IV is the history and eventual proof of the four-color theorem, although it also includes a discussion of generalizations involving coloring maps on surfaces of higher genus. The final chapter gives a number of measurements of a graph\u27s closeness to planarity, including the concepts of crossing number, thickness, splitting number, and coarseness. The chapter conclused with a discussion of two other coloring problems - Heawood\u27s empire problem and Ringel\u27s earth-moon problem

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

Surfaces, Tree-Width, Clique-Minors, and Partitions

In 1971, Chartrand, Geller, and Hedetniemi conjectured that the edge set of a planar graph may be partitioned into two subsets, each of which induces an outerplanar graph. Some partial results towards this conjecture are presented. One such result, in which a planar graph may be thus edge partitioned into two series-parallel graphs, has nice generalizations for graphs embedded onto an arbitrary surface and graphs with no large clique-minor. Several open questions are raised. © 2000 Academic Press

Graph Theory

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