Acrylic improper colorings of graphs

In this article, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph is a vertex-coloring in which adjacent vertices are allowed to have the same color, but each color class V_i satisfies some condition depending on i. Such a coloring is acyclic if there are no alternating 2-colored cycles. We prove that every outerplanar graph can be acyclically 2-colored in such a way that each monochromatic subgraph has degree at most five and that this result is best possible. For planar graphs, we prove some negative results and state some open problems. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 97–107, 1999     

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 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 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 edge coloring of subcubic graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and it is denoted by a^′(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors.     

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 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-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.     

Acyclic improper colouring of graphs with maximum degree 4

A k-colouring(not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given(hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.     

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 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. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:169‐176, 2011     

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 colorings of graphs with bounded degree

A k-coloring (not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colors i and j the subgraph induced by the edges whose endpoints have colors i and j is acyclic. We consider some generalized acyclic k-colorings, namely, we require that each color class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic 5-coloring such that each color class induces an acyclic graph with maximum degree at most 4. We prove that the problem of deciding whether a graph G has an acyclic 2-coloring in which each color class induces a graph with maximum degree at most 3 is NP-complete, even for graphs with maximum degree 5. We also give a linear-time algorithm for an acyclic t-improper coloring of any graph with maximum degree d assuming that the number of colors is large enough.     

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.     

Every toroidal graph is acyclically 8-choosable

A proper coloring of a graphG is acyclic if G contains no 2-colored cycle.A graph G is acyclically L-list colorable if for a given list assignment L={L(v):v∈V(G)},there exists a proper acyclic coloringφof G such thatφ(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 acyclically k-choosable.In this article,we prove that every toroidal graph is acyclically 8-choosable.     

Linear choosability of graphs

A proper vertex coloring of a non-oriented graph G is linear if the graph induced by the vertices of any two color classes is a forest of paths. A graph G is linearly L-list colorable if for a given list assignment L={L(v):v∈V(G)}, there exists a linear coloring c of G such that c(v)∈L(v) for all v∈V(G). If G is linearly L-list colorable for any list assignment with |L(v)|?k for all v∈V(G), then G is said to be linearly k-choosable. In this paper, we investigate the linear choosability for some families of graphs: graphs with small maximum degree, with given maximum average degree, outerplanar and planar graphs. Moreover, we prove that deciding whether a bipartite subcubic planar graph is linearly 3-colorable is an NP-complete problem.     

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.     

Circular list colorings of some graphs

The circular list coloring is a circular version of list colorings of graphs. Let χinc,l denote the circular choosability(or the circular list chromatic number). In this paper, the circular choosability of outer planar graphs and odd wheel is discussed.     

Acyclic 6-choosability of planar graphs without adjacent short cycles

A proper vertex coloring of a graph G is acyclic if G contains no bicolored cycles.Given a list assignment L={L(v)|v∈V}of G,we say that G is acyclically L-colorable if there exists a proper acyclic coloringπof G such thatπ(v)∈L(v)for all v∈V.If G is acyclically L-colorable for any list assignment L with|L(v)|k for all v∈V(G),then G is acyclically k-choosable.In this paper,we prove that every planar graph G is acyclically 6-choosable if G does not contain 4-cycles adjacent to i-cycles for each i∈{3,4,5,6}.This improves the result by Wang and Chen(2009).     

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.     

Sufficient conditions for a graph to be edge-colorable with maximum degree colors

We study properties of graphs related to the existence of certain vertex and edge partitions. These properties give sufficient conditions for a graph to be Class 1 (i.e., edge-colorable with maximum degree colors). We apply these conditions for solving the classification problem for graphs with acyclic core (the subgraph induced by the maximum degree vertices), and for subclasses of join graphs and cobipartite graphs.