Random graphs with few disjoint cycles

The classical Erd\H{o}s-P\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G-B has no cycles. We show that, amongst all such graphs on vertex set {1,..,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class; concerning uniqueness of the blocker, connectivity, chromatic number and clique number. A key step in the proof of the main theorem is to show that there must be a blocker as in the Erd\H{o}s-P\'{o}sa theorem with the extra `redundancy' property that B-v is still a blocker for all but at most k vertices v in B

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. IV. Chromatic polynomial with cyclic boundary conditions

We study the chromatic polynomial P_G(q) for m \times n square- and triangular-lattice strips of widths 2\leq m \leq 8 with cyclic boundary conditions. This polynomial gives the zero-temperature limit of the partition function for the antiferromagnetic q-state Potts model defined on the lattice G. We show how to construct the transfer matrix in the Fortuin--Kasteleyn representation for such lattices and obtain the accumulation sets of chromatic zeros in the complex q-plane in the limit n\to\infty. We find that the different phases that appear in this model can be characterized by a topological parameter. We also compute the bulk and surface free energies and the central charge.Comment: 55 pages (LaTeX2e). Includes tex file, three sty files, and 22 Postscript figures. Also included are Mathematica files transfer4_sq.m and transfer4_tri.m. Journal versio

Functional evidence for cone-specific connectivity in the human retina

NoPhysiological studies of colour vision have not yet resolved the controversial issue of how chromatic opponency is constructed at a neuronal level. Two competing theories, the cone-selective hypothesis and the random-wiring hypothesis, are currently equivocal to the architecture of the primate retina. In central vision, both schemes are capable of producing colour opponency due to the fact that receptive field centres receive input from a single bipolar cell ¿ the so called `private line arrangement¿. However, in peripheral vision this single-cone input to the receptive field centre is lost, so that any random cone connectivity would result in a predictable reduction in the quality of colour vision. Behavioural studies thus far have indeed suggested a selective loss of chromatic sensitivity in peripheral vision. We investigated chromatic sensitivity as a function of eccentricity for the cardinal chromatic (L/M and S/(L + M)) and achromatic (L + M) pathways, adopting stimulus size as the critical variable. Results show that performance can be equated across the visual field simply by a change of scale (size). In other words, there exists no qualitative loss of chromatic sensitivity across the visual field. Critically, however, the quantitative nature of size dependency for each of the cardinal chromatic and achromatic mechanisms is very specific, reinforcing their independence in terms of anatomy and genetics. Our data provide clear evidence for a physiological model of primate colour vision that retains chromatic quality in peripheral vision, thus supporting the cone-selective hypothesis

Phase diagram of the chromatic polynomial on a torus

We study the zero-temperature partition function of the Potts antiferromagnet (i.e., the chromatic polynomial) on a torus using a transfer-matrix approach. We consider square- and triangular-lattice strips with fixed width L, arbitrary length N, and fully periodic boundary conditions. On the mathematical side, we obtain exact expressions for the chromatic polynomial of widths L=5,6,7 for the square and triangular lattices. On the physical side, we obtain the exact ``phase diagrams'' for these strips of width L and infinite length, and from these results we extract useful information about the infinite-volume phase diagram of this model: in particular, the number and position of the different phases.Comment: 72 pages (LaTeX2e). Includes tex file, three sty files, and 26 Postscript figures. Also included are Mathematica files transfer6_sq.m and transfer6_tri.m. Final version to appear in Nucl. Phys.