Few Long Lists for Edge Choosability of Planar Cubic Graphs

It is known that every loopless cubic graph is 4-edge choosable. We prove the following strengthened result. Let G be a planar cubic graph having b cut-edges. There exists a set F of at most 5b/2 edges of G with the following property. For any function L which assigns to each edge of F a set of 4 colours and which assigns to each edge in E(G)-F a set of 3 colours, the graph G has a proper edge colouring where the colour of each edge e belongs to L(e).Comment: 14 pages, 1 figur

A quadratic lower bound for subset sums

Let A be a finite nonempty subset of an additive abelian group G, and let \Sigma(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |\Sigma(A)| >= |H| + 1/64 |A H|^2 where H is the stabilizer of \Sigma(A). Our result implies that \Sigma(A) = Z/nZ for every set A of units of Z/nZ with |A| >= 8 \sqrt{n}. This consequence was first proved by Erd\H{o}s and Heilbronn for n prime, and by Vu (with a weaker constant) for general n.Comment: 12 page

There are only a finite number of excluded minors for the class of bicircular matroids

We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We show that if $N$ is an excluded minor of rank at least ten, then $N$ is quasi-graphic. Several small excluded minors are quasi-graphic. Using biased-graphic representations, we find that $N$ already contains one of these. We also provide an upper bound, in terms of rank, on the number of elements in an excluded minor, so the result follows.Comment: Added an appendix describing all known excluded minors. Added Gordon Royle as author. Some proofs revised and correcte

Cayley sum graphs and eigenvalues of (3,6)(3,6)-fullerenes

We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, "(3,6)-fullerenes", have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form $\{\lambda,-\lambda\}$ except for the four eigenvalues $\{3,-1,-1,-1\}$. We exhibit other families of graphs which are "spectrally nearly bipartite" in this sense. Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum graphs

Circular chromatic number of even-faced projective plane graphs

We derive an exact formula for the circular chromatic number of any graph embeddable on the projective plane in such a way that all of it faces have even length

Edge Disjoint Cycles Through Specified Vertices

Abstract We give a sufficient condition for a simple graph G to have k pairwise edge-disjoint cycles,each of which contains a prescribed set W of vertices. The condition is that the inducedsubgraph G[W] be 2k-connected, and that for any two vertices at distance two in G[W],at least one of the two has degree at least | V (G)|/2 + 2(k- 1) in G. This is a commongeneralization of special cases previously obtained by Bollob'as/Brightwell (where k = 1) andLi (where W = V (G)).A key lemma is of independent interest. Let G be the complement of a bipartite graphwith partite sets X, Y. If G is 2k connected, then G contains k Hamilton cycles which arepairwise edge-disjoint except for edges in G[Y]