||Abstract: This thesis studies five kinds of graph colorings: the circular coloring,|
the total coloring, the (d; 1)-total labeling, the circular (r; 1)-total labeling, and the
acyclic list coloring.
We give upper bounds on the circular chromatic number of graphs with small
maximum average degree, mad for short. It is proved that if mad(G)<22=9 then G
has a 11=4-circular coloring, if mad(G) < 5=2 then G has a 14=5-circular coloring.
A conjecture by Behzad and Vizing implies that Δ+2 colors are always sufficient
for a total coloring of graphs with maximum degree Δ. The only open case for planar graphs is for Δ = 6. Let G be a planar in which no vertex is contained in cycles of all lengths between 3 and 8. If Δ(G) = 6, then G is total 8-colorable. If Δ(G) = 8, then G is total 9-colorable.
Havet and Yu  conjectured that every subcubic graph G ̸=K4 has (2; 1)-total
number at most 5. We confirm the conjecture for graphs with maximum average
degree less than 7=3 and for flower snarks.
We introduce the circular (r; 1)-total labeling. As a relaxation of the aforementioned
conjecture, we conjecture that every subcubic graph has circular (2; 1)-total number at most 7. We confirm the conjecture for graphs with maximum average degree less than 5=2.
We prove that every planar graph with no cycles of lengths 4, 7 and 8 is acyclically
4-choosable. Combined with recent results, this implies that every planar
graph with no cycles of length 4;k; l with 5 6 k < l 6 8 is acyclically 4-choosable.