A theorem on integer flows on cartesian products of graphs

It is shown that the Cartesian product of two nontrivial connected graphs admits a nowhere‐zero 4‐flow. If both factors are bipartite, then the product admits a nowhere‐zero 3‐flow. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 93–98, 2003     

Investigation of nematic order by coherent neutron scattering

The mathematical relationship between the orientational order parameters and the coherent neutron scattering cross section for a nematic liquid crystal is given. For deuterated para-azoxyanisole the single-molecule part of the cross section is evaluated within the meanfield approximation and combined with experimental results to give information about molecular orientational order in terms of P ₂, P ₄ and P ₆. Both P ₂ and P ₄ are found necessary for describing the molecular order. Discrepancies between experimental and theoretical results are interpreted as possibly reflecting the inadequacy of the meanfield theory of Maier and Saupe.     

Orientational fluctuations in a nematic liquid crystal as seen by neutron scattering

Abstract

Real time neutron scattering is used in the study of the slow orientational fluctuations of the director in a nematic sample. A statistical analysis of the observed time series gives the Hurst exponent H and β exponent of the frequency power spectrum that satisfy the scaling relationship β = 2H + 1. In the nematic phase, but not in the solid and in the isotropic liquid phases, the exponent values are those expected for a self-organized critical state. When a magnetic field, of the order of the Freedericksz field is applied, the nematic sample is observed to display persistent oscillations of the director. We confront this observation with theoretical predictions.     

T‐joins intersecting small edge‐cuts in graphs

In an earlier paper 3 , we studied cycles in graphs that intersect all edge‐cuts of prescribed sizes. Passing to a more general setting, we examine the existence of T‐joins in grafts that intersect all edge‐cuts whose size is in a given set A ?{1,2,3}. In particular, we characterize all the contraction‐minimal grafts admitting no T‐joins that intersect all edge‐cuts of size 1 and 2. We also show that every 3‐edge‐connected graft admits a T‐join intersecting all 3‐edge‐cuts. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 64–71, 2007     

Real time observations by neutron scattering of fluctuations near the Rayleigh-Bénard instability

Using long time-series counting in a neutron scattering experiment, hydrodynamic fluctuations were studied near the Rayleigh-Bénard instability in para-azoxyanisole. Critical enhancement was observed and the coupling to inner orientational fluctuations of the nematic liquid crystal was demonstrated through the response of the signal to an external magnetic field. The similarity of the experimental curves to those of critical scattering in equilibrium physics lends support to the phase-transition analogy of hydrodynamic instabilities.Work performed in partial fullfilment of the cand.real.-degree at the Institute of Physics, University of Oslo     

Colorings Of Plane Graphs With No Rainbow Faces

We prove that each n-vertex plane graph with girth g≥4 admits a vertex coloring with at least ⌈n/2⌉+1 colors with no rainbow face, i.e., a face in which all vertices receive distinct colors. This proves a conjecture of Ramamurthi and West. Moreover, we prove for plane graph with girth g≥5 that there is a vertex coloring with at least if g is odd and if g is even. The bounds are tight for all pairs of n and g with g≥4 and n≥5g/2−3. * Supported in part by the Ministry of Science and Technology of Slovenia, Research Project Z1-3129 and by a postdoctoral fellowship of PIMS. ** Institute for Theoretical Computer Science is supported by Ministry of Education of CzechR epublic as project LN00A056.     

Backbone colorings of graphs with bounded degree

We study backbone colorings, a variation on classical vertex colorings: Given a graph G and a subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex k-coloring of G in which the colors assigned to adjacent vertices in H differ by at least 2. The minimal k∈N for which such a coloring exists is called the backbone chromatic number of G. We show that for a graph G of maximum degree Δ where the backbone graph is a d-degenerated subgraph of G, the backbone chromatic number is at most Δ+d+1 and moreover, in the case when the backbone graph being a matching we prove that the backbone chromatic number is at most Δ+1. We also present examples where these bounds are attained.Finally, the asymptotic behavior of the backbone chromatic number is studied regarding the degrees of G and H. We prove for any sparse graph G that if the maximum degree of a backbone graph is small compared to the maximum degree of G, then the backbone chromatic number is at most .     

Non‐rainbow colorings of 3‐, 4‐ and 5‐connected plane graphs

We study vertex‐colorings of plane graphs that do not contain a rainbow face, i.e., a face with vertices of mutually distinct colors. If G is a 3 ‐connected plane graph with n vertices, then the number of colors in such a coloring does not exceed . If G is 4 ‐connected, then the number of colors is at most , and for n≡3(mod8), it is at most . Finally, if G is 5 ‐connected, then the number of colors is at most . The bounds for 3 ‐connected and 4 ‐connected plane graphs are the best possible as we exhibit constructions of graphs with colorings matching the bounds. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 129–145, 2010