124 research outputs found
is a Bound on the Adjacent Vertex Distinguishing Edge Chromatic Number
An adjacent vertex distinguishing edge-coloring or an \avd-coloring of a
simple graph is a proper edge-coloring of such that no pair of adjacent
vertices meets the same set of colors. We prove that every graph with maximum
degree and with no isolated edges has an \avd-coloring with at most
colors, provided that
Random cubic graphs are not homomorphic to the cycle of size 7
We prove that a random cubic graph almost surely is not homomorphic to a
cycle of size 7. This implies that there exist cubic graphs of arbitrarily high
girth with no homomorphisms to the cycle of size 7
Limits of Boolean Functions on F_p^n
We study sequences of functions of the form F_p^n -> {0,1} for varying n, and
define a notion of convergence based on the induced distributions from
restricting the functions to a random affine subspace. Using a decomposition
theorem and a recently proven equi-distribution theorem from higher order
Fourier analysis, we prove that the limits of such convergent sequences can be
represented by certain measurable functions. We are also able to show that
every such limit object arises as the limit of some sequence of functions.
These results are in the spirit of similar results which have been developed
for limits of graph sequences. A more general, albeit substantially more
sophisticated, limit object was recently constructed by Szegedy in [Sze10].Comment: 12 page
- …