Almost all 9-regular graphs have a modulo-5 orientation

In 1972 Tutte famously conjectured that every 4-edge-connected graph has a nowhere zero 3-flow; this is known to be equivalent to every 5-regular, 4-edge-connected graph having an edge orientation in which every in-degree is either 1 or 4. Jaeger conjectured a generalization of Tutte's conjecture, namely, that every $4p+1$-regular, $4p$-edge-connected graph has an edge orientation in which every in-degree is either $p$ or $3p+1$. Inspired by the work of Pralat and Wormald investigating $p=1$, for $p=2$ we show this holds asymptotically almost surely for random 9-regular graphs. It follows that the conjecture holds for almost all 9-regular, 8-edge-connected graphs. These results make use of the technical small subgraph conditioning method.Comment: 18 page

Embedding of Cortical Representations by the Superficial Patch System

Pyramidal cells in layers 2 and 3 of the neocortex of many species collectively form a clustered system of lateral axonal projections (the superficial patch system—Lund JS, Angelucci A, Bressloff PC. 2003. Anatomical substrates for functional columns in macaque monkey primary visual cortex. Cereb Cortex. 13:15-24. or daisy architecture—Douglas RJ, Martin KAC. 2004. Neuronal circuits of the neocortex. Annu Rev Neurosci. 27:419-451.), but the function performed by this general feature of the cortical architecture remains obscure. By comparing the spatial configuration of labeled patches with the configuration of responses to drifting grating stimuli, we found the spatial organizations both of the patch system and of the cortical response to be highly conserved between cat and monkey primary visual cortex. More importantly, the configuration of the superficial patch system is directly reflected in the arrangement of function across monkey primary visual cortex. Our results indicate a close relationship between the structure of the superficial patch system and cortical responses encoding a single value across the surface of visual cortex (self-consistent states). This relationship is consistent with the spontaneous emergence of orientation response-like activity patterns during ongoing cortical activity (Kenet T, Bibitchkov D, Tsodyks M, Grinvald A, Arieli A. 2003. Spontaneously emerging cortical representations of visual attributes. Nature. 425:954-956.). We conclude that the superficial patch system is the physical encoding of self-consistent cortical states, and that a set of concurrently labeled patches participate in a network of mutually consistent representations of cortical inpu

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement