A simpler proof and a generalization of the zero-trees theorem

AbstractZ. Füredi and D. J. Kleitman proved that if an integer weight is assigned to each edge of a complete graph on p + 1 vertices, then some spanning tree has total weight divisible by p. We obtain a simpler proof by generalizing the result to hypergraphs

Circular embeddings of planar graphs in nonspherical surfaces

AbstractWe show that every 3-connected planar graph has a circular embedding in some nonspherical surface. More generally, we characterize those planar graphs that have a 2-representative embedding in some nonspherical surface

A decomposition theorem for binary matroids with no prism minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, $K_5\backslash e$. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac's infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of $F_7$ with itself across a triangle with an element of the triangle deleted; it's rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3-separation in $P_9$. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, $F_7$ and $PG(3, 2)$, respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of $R_{10}$, the unique splitter for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's result identifying the binary internally 4-connected matroids with no prism minor [5]

Disjoint cycles in directed graphs on the torus and the Klein bottle

We give necessary and sufficient conditions for a directed graph embedded on the torus or the Klein bottle to contain pairwise disjoint circuits, each of a given orientation and homotopy, and in a given order. For the Klein bottle, the theorem is new. For the torus, the theorem was proved before by P. D. Seymour. This paper gives a shorter proof of that result. © 1993 by Academic Press, Inc

Directed triangles in directed graphs

AbstractWe show that each directed graph on n vertices, each with indegree and outdegree at least n/t, where t=5−5+1247−215=2.8670975⋯, contains a directed circuit of length at most 3

Solution of two fractional packing problems of Lovasz

AbstractLovász asked whether the following is true for each hypergraph H and natural number k:(∗) if vk (H′) = k · v∗ (H′) holds for each hypergraph H′ arising from H by multiplication of points, then vk(H) = τk(H);(∗∗) if τk(H′) = k · τ∗(H′) holds for each hypergraph H′ arising by removing edges, then τk (H) = vk (H).We prove and generalize assertion (∗) and give a counterexample to (∗∗)

On the odd-minor variant of Hadwiger's conjecture

A {\it $K_l$ -expansion} consists of $l$ vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion {\it odd} if its vertices can be two-coloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every $l$, if a graph contains no odd $K_l$ -expansion then its chromatic number is $O(l \sqrt{\log l})$. In doing so, we obtain a characterization of graphs which contain no odd $K_l$ -expansion which is of independent interest. We also prove that given a graph and a subset $S$ of its vertex set, either there are $k$ vertex-disjoint odd paths with endpoints in $S$, or there is a set X of at most $2k − 2$ vertices such that every odd path with both ends in $S$ contains a vertex in $X$. Finally, we discuss the algorithmic implications of these results