Δ+300\Delta+300 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 $G$ is a proper edge-coloring of $G$ such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree $\Delta$ and with no isolated edges has an \avd-coloring with at most $\Delta+300$ colors, provided that $\Delta >10^{20}$

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