In an earlier paper, we proved (with Chudnovsky and Spirkl) that for all integers κ, ` ≥ 0, every graph with clique number at most κ and sufficiently large chromatic number has an odd hole of length at least `. Here we prove a strengthening, that for all integers κ, ` ≥ 0, every graph with clique number at most κ and sufficiently large chromatic number has a hole of every possible length modulo `.

We give a decomposition theorem for signed graphs whose frame matroids are binary and a decomposition theorem for signed graphs whose frame matroids are quaternary.

Binary strings of length n can be added and permuted. Of special interest is the permutation that cyclicly shifts a binary string to the right. For example, when n = 5, then cyclicly shifting 10110 to the right yields 01011. Obviously if we cyclicly shift a string of length n to the right n times to get back what we started with. Addition of the strings is done component-wise modulo 2. For inst...

We present elements of H(C×C, K2) for certain specific curves C. The image of the element under the boundary map arising from the localization sequence of K-theory is the graph of frobenius endomorphism of the reduction of the curve modulo 3. §

A cycle in a graph is a set of edges that covers each vertex an even number of times. An even cycle is cycle of even cardinality. A co cycle is a collection of edges that intersects each cycle in an even number of edges. A coeven is a co cycle or the complement of a cocycle. A bieven is a collection of edges that is both an even cycle and a coeven. The even cycles, coevens, and bievens each for...

To study the block structure of a connected graph G = (V,E), we introduce two algebraic approaches that reflect this structure: a binary operation + called a leap operation and a ternary relation L called a leap system, both on a finite, nonempty set V . These algebraic structures are easily studied by considering their underlying graphs, which turn out to be block graphs. Conversely, we define...

This paper proposes a data model incorporating the concepts of directed graphs, recursive graphs, and hypergraphs in order to represent the contents of multimedia data. In the proposed data model, an instance is represented with a directed recursive hypergraph. This graph is called an instance graph. A collection of instance graphs is managed as a graph named a collection graph. A shape graph, ...