Anna O. Ivanova

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

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 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 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.

5-Stars of Low Weight in Normal Plane Maps with Minimum Degree 5

Abstract It is known that there are normal plane maps M5 with minimum degree 5 such that the minimum degree-sum w(S5) of 5-stars at 5-vertices is arbitrarily large. In 1940, Lebesgue showed that if an M5 has no 4-stars of cyclic type (5, 6, 6, 5) centered at 5-vertices, then w(S5) ≤ 68. We improve this bound of 68 to 55 and give a construction of a (5, 6, 6, 5)-free M5 with w(S5) = 48

List 2-distance (Δ + 1)-coloring of planar graphs with given girth

Some sufficient conditions (in terms of the girth and maximum degree) are given for the list 2-distance chromatic number of a planar graph with maximum degree Δ to be equal to Δ + 1.

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

Abstract In 1940, Lebesgue proved that every 3-polytope with minimum degree at least 4 contains a 3-face for which the set of degrees of its vertices is majorized by one of the following sequences: ( 4 , 4 , ∞ ) , ( 4 , 5 , 19 ) , ( 4 , 6 , 11 ) , ( 4 , 7 , 9 ) , ( 5 , 5 , 9 ) , ( 5 , 6 , 7 ) . Borodin (2002) strengthened this to ( 4 , 4 , ∞ ) , ( 4 , 5 , 17 ) , ( 4 , 6 , 11 ) , ( 4 , 7 , 8 ) , ( 5 , 5 , 8 ) , ( 5 , 6 , 6 ) . We obtain the following description of 3-faces in normal plane maps with minimum degree at least 4 (in particular, it holds for 3-polytopes) in which every parameter is best possible and is attained independently of the others: ( 4 , 4 , ∞ ) , ( 4 , 5 , 14 ) , ( 4 , 6 , 10 ) , ( 4 , 7 , 7 ) , ( 5 , 5 , 7 ) , ( 5 , 6 , 6 ) .

Acyclic 3-choosability of sparse graphs with girth at least 7

Every planar graph is known to be acyclically 7-choosable and is conjectured to be acyclically 5-choosable (Borodin et al. 2002 [4]). This conjecture if proved would imply both Borodins acyclic 5-color theorem (1979) and Thomassens 5-choosability theorem (1994). However, as yet it has been verified only for several restricted classes of graphs. Some sufficient conditions have also been obtained for a planar graph to be acyclically 4- and 3-choosable. We prove that each planar graph of girth at least 7 is acyclically 3-choosable. This is a common strengthening of the facts that such a graph is acyclically 3-colorable (Borodin et al., 1999 [10]) and that a planar graph of girth at least 8 is acyclically 3-choosable (Montassier et al., 2006 [19]). More generally, we prove that every graph with girth at least 7 and maximum average degree less than 145 is acyclically 3-choosable.

Oriented 5-Coloring of Sparse Plane Graphs

An oriented k-coloring of an oriented graph H is defined to be an oriented homomorphism of H into a k-vertex tournament. It is proved that every orientation of a graph with girth at least 5 and maximum average degree over all subgraphs less than 12/5 has an oriented 5-coloring. As a consequence, each orientation of a plane or projective plane graph with girth at least 12 has an oriented 5-coloring.

Acyclic 5-choosability of planar graphs without adjacent short cycles

The conjecture on acyclic 5-choosability of planar graphs [Borodin et al., 2002] as yet has been verified only for several restricted classes of graphs. None of these classes allows 4-cycles. We prove that a planar graph is acyclically 5-choosable if it does not contain an i-cycle adjacent to a j-cycle where 3⩽j⩽5 if i = 3 and 4⩽j⩽6 if i = 4. This result absorbs most of the previous work in this direction.

Acyclic 4-choosability of planar graphs with neither 4-cycles nor triangular 6-cycles

Every planar graph is known to be acyclically 7-choosable and is conjectured to be acyclically 5-choosable (Borodin et al. 2002) [7]. This conjecture if proved would imply both Borodins acyclic 5-color theorem (1979) and Thomassens 5-choosability theorem (1994). However, as yet it has been verified only for several restricted classes of graphs. Some sufficient conditions are also obtained for a planar graph to be acyclically 4-choosable and 3-choosable. In particular, acyclic 4-choosability was proved for the following planar graphs: without 3-cycles and 4-cycles (Montassier, 2006 [23]), without 4-cycles, 5-cycles and 6-cycles (Montassier et al. 2006 [24]), and either without 4-cycles, 6-cycles and 7-cycles, or without 4-cycles, 6-cycles and 8-cycles (Chen et al. 2009 [14]). In this paper it is proved that each planar graph with neither 4-cycles nor 6-cycles adjacent to a triangle is acyclically 4-choosable, which covers these four results.