Planar Graphs Without 4-Cycles Are (6, 6)-Colorable

It has been shown that there is a planar graph without 3-cycles which is not (Δ1,Δ2)-colorable for any given Δ1,Δ2. This inspires many research to obtain sufficient conditions for planar graphs without 4-cycles and other cycles to be (Δ1,Δ2)-colorable. For example, planar graphs without k-cycles and 4-cycles are (Δ1,Δ2)-colorable for each k∈{3,5} and (Δ1,Δ2)={(2,6),(3,4)}. In this work, we study the values of Δ1 and Δ2 that make a planar graph without only 4-cycles (Δ1,Δ2)-colorable. We begin with Δ1=6 and Δ2=6. Planar graphs without 4-cycles are (6,6)-colorable. Two colors in a (Δ1,Δ2)-coloring, where Δ1=Δ2 are switchable, thus reflect the symmetry of the resulting coloring. Furthermore, some proof techniques are using the symmetry of these two colors.