系列平行图的围长和圆可选性

本文讨论了系列平行图的圆可选性．令x_(c,l)表示圆可选性(或圆列表着色数)．本文证明了围长至少是4n 1的系列平行图的圆可选性至多为2 1/n．     

ON 3-CHOOSABIL ITY OF PL ANE GRAPHSON3 -CHOOSABIL ITY OF PL ANE GRAPHS WITHOUT 6-,7-AND 9-CYCLES

§ 1　 IntroductionAll graphs considered in this paper are finite,simple plane graphs.G=(V,E,F)denotes a plane graph,with V,E and F being the set of vertices,edges and faces of G,respectively.Two vertices u and v are adjacent,denoted by uv∈E,if there is an edge in Ejoining them.A vertex u is incident with an edge e if u is an endvertex of e.Two faces aresaid to be adjacent if they share a common edge.We use b(f) to denote the boundary of aface f.A face is incident with all vertices and e…     

The Erd?s-Pósa property for vertex- and edge-disjoint odd cycles in graphs on orientable surfaces

We prove that for any orientable surface S and any non-negative integer k, there exists an integer f_S(k) such that every graph G embeddable in S has either k vertex-disjoint odd cycles or a vertex set A of cardinality at most f_S(k) such that G-A is bipartite. Such a property is called the Erd?s-Pósa property for odd cycles. We also show its edge version. As Reed [Mangoes and blueberries, Combinatorica 19 (1999) 267-296] pointed out, the Erd?s-Pósa property for odd cycles do not hold for all non-orientable surfaces.     

Volume entropy,weighted girths and stable balls on graphs

We prove new isoperimetric inequalities on graphs involving quantities linked with concepts from differential geometry. First, we bound from above the product of the volume entropy (defined as the log of the exponential growth rate of the universal cover) and the girth of weighted graphs in terms of their cyclomatic number. In a second part, we study a natural polyhedron associated to a weighted graph: the stable ball. In particular, we relate the volume of this polyhedron, the weight of the graph and its cyclomatic number. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 291–305, 2007     

A contribution to queens graphs: A substitution method

A graph G is a queens graph if the vertices of G can be mapped to queens on the chessboard such that two vertices are adjacent if and only if the corresponding queens attack each other, i.e. they are in horizontal, vertical or diagonal position.We prove a conjecture of Beineke, Broere and Henning that the Cartesian product of an odd cycle and a path is a queens graph. We show that the same does not hold for two odd cycles. The representation of the Cartesian product of an odd cycle and an even cycle remains an open problem.We also prove constructively that any finite subgraph of the rectangular grid or the hexagonal grid is a queens graph.Using a small computer search we solve another conjecture of the authors mentioned above, saying that K_3,4 minus an edge is a minimal non-queens graph.     

Tough Ramsey Graphs Without Short Cycles

A graph G is t-tough if any induced subgraph of it with x > 1 connected components is obtained from G by deleting at least tx vertices. It is shown that for every t and g there are t-tough graphs of girth strictly greater than g. This strengthens a recent result of Bauer, van den Heuvel and Schmeichel who proved the above for g = 3, and hence disproves in a strong sense a conjecture of Chvátal that there exists an absolute constant t ₀ so that every t ₀-tough graph is pancyclic. The proof is by an explicit construction based on the tight relationship between the spectral properties of a regular graph and its expansion properties. A similar technique provides a simple construction of triangle-free graphs with independence number m on (m ^4/3) vertices, improving previously known explicit constructions by Erdös and by Chung, Cleve and Dagum.     

Bodies with similar projections

Aleksandrov's projection theorem characterizes centrally symmetric convex bodies by the measures of their orthogonal projections on lower dimensional subspaces. A general result proved here concerning the mixed volumes of projections of a collection of convex bodies has the following corollary. If is a convex body in whose projections on -dimensional subspaces have the same -dimensional volume as the projections of a centrally symmetric convex body , then the Quermassintegrals satisfy , for , with equality, for any , if and only if is a translate of . The case where is centrally symmetric gives Aleksandrov's projection theorem.

     

The equivariant family index theorem in odd dimensions

In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed's odd dimensional index formula. Then we extend this theorem to the noncommutative geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.     

The homomorphism threshold of -free graphs

We determine the structure of -free graphs with n vertices and minimum degree larger than : such graphs are homomorphic to the graph obtained from a -cycle by adding all chords of length , for some k. This answers a question of Messuti and Schacht. We deduce that the homomorphism threshold of -free graphs is 1/5, thus answering a question of Oberkampf and Schacht.