Oleg V. Borodin

Colorings of plane graphs: A survey

Abstract After a brief historical account, a few simple structural theorems about plane graphs useful for coloring are stated, and two simple applications of discharging are given. Afterwards, the following types of proper colorings of plane graphs are discussed, both in their classical and choosability (list coloring) versions: simultaneous colorings of vertices, edges, and faces (in all possible combinations, including total coloring), edge-coloring, cyclic coloring (all vertices in any small face have different colors), 3-coloring, acyclic coloring (no 2-colored cycles), oriented coloring (homomorphism of directed graphs to small tournaments), a special case of circular coloring (the colors are points of a small cycle, and the colors of any two adjacent vertices must be nearly opposite on this cycle), 2-distance coloring (no 2-colored paths on three vertices), and star coloring (no 2-colored paths on four vertices). The only improper coloring discussed is injective coloring (any two vertices having a common neighbor should have distinct colors).

Total colorings of planar graphs with large maximum degree

It is proved that a planar graph with maximum degree Δ ≥ 11 has total (vertex-edge) chromatic number

Total Colourings of Planar Graphs with Large Girth

Delta; + 1.

2-distance (Δ+2)-coloring of planar graphs with girth six and Δ≥18

It is proved that ifGis a planar graph with total (vertex?edge) chromatic number ??, maximum degree ? and girthg, then ??=?+1 if ??5 andg?5, or ??4 andg?6, or ??3 andg?10. These results hold also for graphs in the projective plane, torus and Klein bottle.

On cyclic colorings and their generalizations

It was proved in [Z. Dvorak, D. Kral, P. Nejedly@?, R. Skrekovski, Coloring squares of planar graphs with girth six, European J. Combin. 29 (4) (2008) 838-849] that every planar graph with girth g>=6 and maximum degree @D>=8821 is 2-distance (@D+2)-colorable. We prove that every planar graph with g>=6 and @D>=18 is 2-distance (@D+2)-colorable.

Describing (d−2)-stars at d-vertices, d≤5, in normal plane maps

Abstract A cyclic coloring is a vertex coloring such that vertices in a face receive different colors. Let Δ be the maximum face degree of a graph. This article shows that plane graphs have cyclic 9 5 Δ -colorings, improving results of Ore and Plummer, and of Borodin. The result is mainly a corollary of a best-possible upper bound on the minimum cyclic degree of a vertex of a plane graph in terms of its maximum face degree. The proof also yields results on the projective plane, as well as for d-diagonal colorings. Also, it is shown that plane graphs with Δ=5 have cyclic 8-colorings. This result and also the 9 5 Δ result are not necessarily best possible.

On 1-improper 2-coloring of sparse graphs

Abstract We prove that every normal plane map has a ( 3 , 1 0 − ) -edge, or a ( 5 − , 4 , 9 − ) -path, or a ( 6 , 4 , 8 − ) -path, or a ( 7 , 4 , 7 ) -path, or a ( 5 ; 4 , 5 , 5 ) -star, or a ( 5 ; 5 , b , c ) -star with 5 ≤ b ≤ 6 and 5 ≤ c ≤ 7 , or a ( 5 ; 6 , 6 , 6 ) -star. Moreover, none of the above options can be strengthened or dropped. In particular, this extends or strengthens several known results and disproves a related conjecture of Harant and Jendrol’ (2007) [10] .

Describing 3-paths in normal plane maps

Abstract A graph G is ( 1 , 1 ) -colorable if its vertices can be partitioned into subsets V 1 and V 2 such that every vertex in G [ V i ] has degree at most 1 for each i ∈ { 1 , 2 } . We prove that every graph with maximum average degree at most 14 5 is ( 1 , 1 ) -colorable. In particular, it follows that every planar graph with girth at least 7 is ( 1 , 1 ) -colorable. On the other hand, we construct graphs with maximum average degree arbitrarily close to 14 5 (from above) that are not ( 1 , 1 ) -colorable. In fact, we establish the best possible sufficient condition for the ( 1 , 1 ) -colorability of a graph G in terms of the minimum, ρ G , of ρ G ( S ) = 7 | S | − 5 | E ( G [ S ] ) | over all subsets S of V ( G ) . Namely, every graph G with ρ G ≥ 0 is ( 1 , 1 ) -colorable. On the other hand, we construct infinitely many non- ( 1 , 1 ) -colorable graphs G with ρ G = − 1 . This solves a related conjecture of Kurek and Rucinski from 1994.

Describing 4-stars at 5-vertices in normal plane maps with minimum degree 5

Abstract We prove that every normal plane map, as well as every 3-polytope, has a path on three vertices whose degrees are bounded from above by one of the following triplets: ( 3 , 3 , ∞ ) , ( 3 , 4 , 11 ) , ( 3 , 7 , 5 ) , ( 3 , 10 , 4 ) , ( 3 , 15 , 3 ) , ( 4 , 4 , 9 ) , ( 6 , 4 , 8 ) , ( 7 , 4 , 7 ) , and ( 6 , 5 , 6 ) . No parameter of this description can be improved, as shown by appropriate 3-polytopes.

Describing 3-faces in normal plane maps with minimum degree 4

Abstract Lebesgue (1940)xa0 [13] proved that each plane normal map M 5 with minimum degree 5 has a 5-vertex such that the degree-sum (the weight) of its every four neighbors is at most 26. In other words, every M 5 has a 4-star of weight at most 31 centered at a 5-vertex. Borodin–Woodall (1998)xa0 [3] improved this 31 to the tight bound 30. We refine the tightness of Borodin–Woodall’s bound 30 by presenting six M 5 s such that (1) every 4-star at a 5-vertex in them has weight at least 30 and (2) for each of the six possible types ( 5 , 5 , 5 , 10 ) , ( 5 , 5 , 6 , 9 ) , ( 5 , 5 , 7 , 8 ) , ( 5 , 6 , 6 , 8 ) , ( 5 , 6 , 7 , 7 ) , and ( 6 , 6 , 6 , 7 ) of 4-stars with weight 30, the 4-stars of this type at 5-vertices appear in precisely one of these six M 5 s.