Abstract
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.
Highlights
Graph coloring is a very important problem in graph theory
For planar graphs, Vizing [1] proved that any planar graph with Δ ≥ 8 is of class 1 and illustrated that there are class 2 planar graphs with Δ ∈ {2, 3, 4, 5}. erefore, Vizing conjectured that any planar graph with Δ ∈ {6, 7} is of class 1
We present a result concerning the edge chromatic number of planar graphs with Δ 6 which do not contain any adjacent 8-cycles
Summary
Graph coloring is a very important problem in graph theory. Since the four-color problem was first proposed, many other forms of coloring problems have been put forward and extensively studied. (2.3) if the remaining face is a 4-face and δ1(v) 3, there are two cases for each 6-neighbor x of v (2.1) Suppose that the remaining face is an 8+-face and any 3-neighbors and any 5-neighbors of v are not adjacent.
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