Planar Graphs of Maximum Degree 6 and without Adjacent 8-Cycles Are 6-Edge-Colorable

In this paper, by applying the discharging method, we show that if G is a planar graph with a maximum degree of Δ = 6 that does not contain any adjacent 8-cycles, then G is of class 1.

Erdös-Gyárfás Conjecture for Cubic Planar Graphs

In 1995, Paul Erdös and András Gyárfás conjectured that for every graph of minimum degree at least 3, there exists a non-negative integer $m$ such that $G$ contains a simple cycle of length $2^m$. In this paper, we prove that the conjecture holds for 3-connected cubic planar graphs. The proof is long, computer-based in parts, and employs the Discharging Method in a novel way.