The circumference of a graph with no K3, t-minor

It was shown by Chen and Yu that every 3-connected planar graph G contains a cycle of length at least | G |log 3 2, where | G | denotes the number of vertices of G. Thomas made a conjecture in a more general setting: there exists a function β (t) > 0 for t ≥ 3, such that every 3-connected graph G with no K3, t-minor, t ≥ 3, contains a cycle of length at least | G |β (t). We prove that this conjecture is true with β (t) = log8 t t + 1 2. We also show that every 2-connected graph with no K2, t-minor, t ≥ 3, contains a cycle of length at least | G | / tt - 1. © 2006 Elsevier Inc. All rights reserved.preprin

The circumference of a graph with no K3,t-minor, II

The class of graphs with no K3;t-minors, t>=3, contains all planar graphs and plays an important role in graph minor theory. In 1992, Seymour and Thomas conjectured the existence of a function α(t)>0 and a constant β>0, such that every 3-connected n-vertex graph with no K3;t-minors, t>=3, contains a cycle of length at least α(t)nβ. The purpose of this paper is to con¯rm this conjecture with α(t)=(1/2)t(t-1) and β=log1729 2.preprin

Density and Chromatic Index, and Minimum Ranks of Sign Pattern Matrices

Given a (multi)graph, the density is defined by $$\Gamma(G)=\max \Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\}.$$ The {\bf chromatic index} $\chi\u27(G)$ of a graph $G$ is the minimum number of colors that required to color the edges of $G$ such that two adjacent edges receive different colors. It is known that $\chi\u27(G)\geq \Gamma(G)$. The {\bf cover index} $\xi(G)$ of $G$ is the greatest integer $k$ for which there is a coloring of $E$ with $k$ colors such that each vertex of $G$ is incident with at least one edge of each color. A sign pattern is a matrix whose entries are from the set $\{+, -, 0\}$. In part 1, we will generally discuss the connections between the density and the chromatic index. In particular, the Goldberg-Seymour conjecture states that $\chi\u27(G)=\lceil\Gamma(G)\rceil$ if $\chi\u27(G)\u3e\Delta+1$, where $\Delta$ is the maximum degree of $G$. Some open problems are mentioned at the end of part 1. In particular, a dual conjecture to the Goldberg-Seymour conjecture on the cover index is discussed. A proof of the Goldberg-Seymour conjecture is given In part 2. In part 3, we will present a connection between the minimum ranks of sign pattern matrices and point-line configurations