Acyclic 5-choosability of planar graphs without 4-cycles

The conjecture on the acyclic 5-choosability of planar graphs (Borodin et al., 2002) as yet has been verified only for several restricted classes of graphs: those of girth at least 5 (Montassier, Ochem, and Raspaud, 2006), without 4- and 5-cycles or without 4- and 6-cycles (Montassier, Raspaud, and Wang, 2007), with neither 4-cycles nor chordal 6-cycles (Zhang and Xu, 2009), with neither 4- cycles nor two 3-cycles at distance less than 3 (Chen and Wang, 2008), and with neither 4-cycles nor intersecting 3-cycles (Chen and Raspaud, 2010). Wang and Chen (2009) proved that the planar graphs without 4-cycles are acyclically 6-choosable. We prove that a planar graph without 4-cycles is acyclically 5-choosable, which is a common strengthening of all above-mentioned results.     

Acyclic 5-choosability of planar graphs without 4-cycles

A proper vertex coloring of a graph G=(V,E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L-list colorable if for a given list assignment L={L(v):v∈V}, there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L-list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k-choosable. In this paper we prove that every planar graph without 4-cycles and without two 3-cycles at distance less than 3 is acyclically 5-choosable. This improves a result in [M. Montassier, P. Ochem, A. Raspaud, On the acyclic choosability of graphs, J. Graph Theory 51 (2006) 281-300], which says that planar graphs of girth at least 5 are acyclically 5-choosable.     

Acyclic 5-choosability of planar graphs with neither 4-cycles nor chordal 6-cycles

A proper vertex coloring of a graph G=(V,E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L-list colorable if for a given list assignment L={L(v):v∈V}, there exists a proper acyclic coloring ? of G such that ?(v)∈L(v) for all v∈V(G). If G is acyclically L-list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k-choosable. In this paper it is proved that every planar graph with neither 4-cycles nor chordal 6-cycles is acyclically 5-choosable. This generalizes the results of [M. Montassier, A. Raspaud, W. Wang, Acyclic 5-choosability of planar graphs without small cycles, J. Graph Theory 54 (2007) 245-260], and a corollary of [M. Montassier, P. Ochem, A. Raspaud, On the acyclic choosability of graphs, J. Graph Theory 51 (4) (2006) 281-300].     

Acyclic 3-choosability of planar graphs without cycles of length from 4 to 12

Every planar graph is known to be acyclically 7-choosable and is conjectured to be acyclically 5-choosable (O. V. Borodin et al., 2002). This conjecture if proved would imply both Borodin’s acyclic 5-color theorem (1979) and Thomassen’s 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- and 3-colorable. In particular, a planar graph of girth at least 7 is acyclically 3-colorable (O. V. Borodin, A. V. Kostochka and D. R. Woodall, 1999) and acyclically 3-choosable (O. V. Borodin et. al, 2009). A natural measure of sparseness, introduced by Erdős and Steinberg, is the absence of k-cycles, where 4 ≤ k ≤ S. Here, we prove that every planar graph without cycles of length from 4 to 12 is acyclically 3-choosable.     

Acyclic 4-coloring of planar graphs without 4- and 5-cycles

Every planar graph is known to be acyclically 5-colorable (O.V.Borodin, 1976). Some sufficient conditions are also obtained for a planar graph to be acyclically 4- and 3-colorable. In particular, the acyclic 4-colorability was proved for the following planar graphs: without 3- and 4-cycles (O.V.Borodin, A.V. Kostochka, and D.R.Woodall, 1999), without 4-, 5-, and 6-cycles, or without 4-, 5-, and 7-cycles, or without 4-, 5-, and intersecting 3-cycles (M. Montassier, A.Raspaud andW.Wang, 2006), and without 4-, 5-, and 8-cycles (M. Chen and A.Raspaud, 2009). The purpose of this paper is to prove that each planar graph without 4- and 5-cycles is acyclically 4-colorable.     

Acyclic 4-colorability of planar graphs without cycles of length 4 or 6

Every planar graph is known to be acyclically 5-colorable (O. V. Borodin, 1976), which bound is precise. Some sufficient conditions are also obtained for a planar graph to be acyclically 4-colorable. In particular, the acyclic 4-colorabilitywas proved for the following planar graphs: without 3- and 4-cycles (O. V. Borodin, A. V. Kostochka, and D. R. Woodall, 1999), without 4-, 5-, and 6-cycles (M. Montassier, A. Raspaud, and W. Wang, 2006), and either without 4-, 6-, and 7-cycles or without 4-, 6-, and 8-cycles (M. Chen, A. Raspaud, and W. Wang, 2009). In this paper it is proved that each planar graph with neither 4-cycles nor 6-cycles is acyclically 4-colorable.     

A note on the acyclic 3-choosability of some planar graphs

An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G. A list assignment of G is a function L that assigns to each vertex v∈V(G) a list L(v) of available colors. Let G be a graph and L be a list assignment of G. The graph G is acyclically L-list colorable if there exists an acyclic coloring ? of G such that ?(v)∈L(v) for all v∈V(G). If G is acyclically L-list colorable for any list assignment L with |L(v)|≥k for all v∈V(G), then G is said to be acyclically k-choosable. Borodin et al. proved that every planar graph with girth at least 7 is acyclically 3-choosable (Borodin et al., submitted for publication [4]). More recently, Borodin and Ivanova showed that every planar graph without cycles of length 4 to 11 is acyclically 3-choosable (Borodin and Ivanova, submitted for publication [7]). In this note, we connect these two results by a sequence of intermediate sufficient conditions that involve the minimum distance between 3-cycles: we prove that every planar graph with neither cycles of lengths 4 to 7 (resp. to 8, to 9, to 10) nor triangles at distance less than 7 (resp. 5, 3, 2) is acyclically 3-choosable.     

Acyclic 4‐Choosability of Planar Graphs with No 4‐ and 5‐Cycles

Every planar graph is known to be acyclically 7‐choosable and is conjectured to be acyclically 5‐choosable (O. V. Borodin, D. G. Fon‐Der‐Flaass, A. V. Kostochka, E. Sopena, J Graph Theory 40 (2002), 83–90). This conjecture if proved would imply both Borodin's (Discrete Math 25 (1979), 211–236) acyclic 5‐color theorem and Thomassen's (J Combin Theory Ser B 62 (1994), 180–181) 5‐choosability theorem. 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‐ and 3‐choosable. In particular, the acyclic 4‐choosability was proved for the following planar graphs: without 3‐, 4‐, and 5‐cycles (M. Montassier, P. Ochem, and A. Raspaud, J Graph Theory 51 (2006), 281–300), without 4‐, 5‐, and 6‐cycles, or without 4‐, 5‐, and 7‐cycles, or without 4‐, 5‐, and intersecting 3‐cycles (M. Montassier, A. Raspaud, W. Wang, Topics Discrete Math (2006), 473–491), and neither 4‐ and 5‐cycles nor 8‐cycles having a triangular chord (M. Chen and A. Raspaud, Discrete Math. 310(15–16) (2010), 2113–2118). The purpose of this paper is to strengthen these results by proving that each planar graph without 4‐ and 5‐cycles is acyclically 4‐choosable.     

Planar graphs without 3-, 7-, and 8-cycles are 3-choosable

A graph G is k-choosable if every vertex of G can be properly colored whenever every vertex has a list of at least k available colors. Grötzsch’s theorem [4] states that every planar triangle-free graph is 3-colorable. However, Voigt [M. Voigt, A not 3-choosable planar graph without 3-cycles, Discrete Math. 146 (1995) 325-328] gave an example of such a graph that is not 3-choosable, thus Grötzsch’s theorem does not generalize naturally to choosability. We prove that every planar triangle-free graph without 7- and 8-cycles is 3-choosable.     

DP-4-colorability of planar graphs without intersecting 5-cycles

DP-coloring of graphs as a generalization of list coloring was introduced by Dvo?ák and Postle (2018). In this paper, we show that every planar graph without intersecting 5-cycles is DP-4-colorable, which improves the result of Hu and Wu (2017), who proved that every planar graph without intersecting 5-cycles is 4-choosable, and the results of Kim and Ozeki (2018).     

Acyclic 4-choosability of planar graphs

A proper vertex coloring of a graph G=(V,E) is acyclic if G contains no bicolored cycle. Given a list assignment L={L(v)∣v∈V} of G, we say G is acyclically L-list colorable if there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L-list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k-choosable. In this paper we prove that planar graphs without 4, 7, and 8-cycles are acyclically 4-choosable.     

On acyclic edge coloring of planar graphs without intersecting triangles

Let Δ denote the maximum degree of a graph. Fiam?ík first and then Alon et al. again conjectured that every graph is acyclically edge (Δ+2)-colorable. Even for planar graphs, this conjecture remains open. It is known that every triangle-free planar graph is acyclically edge (Δ+5)-colorable. This paper proves that every planar graph without intersecting triangles is acyclically edge (Δ+4)-colorable.     

A note on the Three Color Problem on planar graphs without 4- and 5-cycles and without ext-triangular 7-cycles

Steinberg conjectured in 1976 that every planar graph with no cycles of length four or five is 3-colorable. This conjecture is disproved by constructing a planar graph with no cycles of length four or five but intersecting triangles. Jin et al. proved that plane graphs without 4- and 5-cycles and without ext-triangular 7-cycles are 3-colorable [SIAM J. Discrete Math. 31 (3) (2017) 1836–1847]. In this paper, we point out a mistake of their proof and give an improved proof.     

Planar graphs without triangles adjacent to cycles of length from 4 to 7 are 3-colorable

It is known that planar graphs without cycles of length from 4 to 7 are 3-colorable (Borodin et al., 2005) [13] and that planar graphs in which no triangles have common edges with cycles of length from 4 to 9 are 3-colorable (Borodin et al., 2006) [11]. We give a common extension of these results by proving that every planar graph in which no triangles have common edges with k-cycles, where k∈{4,5,7} (or, which is equivalent, with cycles of length 3, 5 and 7), is 3-colorable.     

Acyclic Edge Coloring of Planar Graphs Without Small Cycles

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a′(G) ≤ Δ(G) + 2 for any graphs. In this paper, it is shown that the conjecture holds for planar graphs without 4- and 5-cycles or without 4- and 6-cycles.     

Acyclic improper choosability of graphs

We consider improper colorings (sometimes called generalized, defective or relaxed colorings) in which every color class has a bounded degree. We propose a natural extension of improper colorings: acyclic improper choosability. We prove that subcubic graphs are acyclically (3, 1)*-choosable (i.e. they are acyclically 3-choosable with color classes of maximum degree one). Using a linear time algorithm, we also prove that outerplanar graphs are acyclically (2, 5)*-choosable (i.e. they are acyclically 2-choosable with color classes of maximum degree five). Both results are optimal. We finally prove that acyclic choosability and acyclic improper choosability of planar graphs are equivalent notions.     

A non-3-choosable planar graph without cycles of length 4 and 5

Steinberg's question from 1975 whether every planar graph without 4- and 5-cycles is 3-colorable is still open. In this paper the analogous question for 3-choosability of such graphs is answered to the negative.     

3-list-coloring planar graphs of girth 4

In Thomassen (1995) [4], Thomassen proved that planar graphs of girth at least 5 are 3-choosable. In Li (2009) [3], Li improved Thomassen’s result by proving that planar graphs of girth 4 with no 4-cycle sharing a vertex with another 4- or 5-cycle are 3-choosable. In this paper, we prove that planar graphs of girth 4 with no 4-cycle sharing an edge with another 4- or 5-cycle are 3-choosable. It is clear that our result strengthens Li’s result.