Edge choosability of planar graphs without 5-cycles with a chord

Let G be a plane graph having no 5-cycles with a chord. If either Δ≥6, or Δ=5 and G contains no 4-cycles with a chord or no 6-cycles with a chord, then G is edge-(Δ+1)-choosable, where Δ denotes the maximum degree of G.     

Edge choosability and total choosability of planar graphs with no 3-cycles adjacent 4-cycles

Two cycles are said to be adjacent if they share a common edge. Let G be a planar graph without triangles adjacent 4-cycles. We prove that if Δ(G)≥6, and and if Δ(G)≥8, where and denote the list edge chromatic number and list total chromatic number of G, respectively.     

On the 3-colorability of planar graphs without 4-, 7- and 9-cycles

In this paper, we mainly prove that planar graphs without 4-, 7- and 9-cycles are 3-colorable.     

Total choosability of planar graphs with maximum degree 4

Let G be a planar graph with maximum degree 4. It is known that G is 8-totally choosable. It has been recently proved that if G has girth g?6, then G is 5-totally choosable. In this note we improve the first result by showing that G is 7-totally choosable and complete the latter one by showing that G is 6-totally choosable if G has girth at least 5.     

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.     

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 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 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.     

Defective 2-colorings of planar graphs without 4-cycles and 5-cycles

A 2-coloring is a coloring of vertices of a graph with colors 1 and 2. Define $V_i?\{v\in V\left(G\right):c\left(v\right)=i\}$ for $i=1$ and $2.$ We say that $G$ is $(d_1,d_2)$-colorable if $G$ has a 2-coloring such that $V_i$ is an empty set or the induced subgraph $G\left[V_i\right]$ has the maximum degree at most $d_i$ for $i=1$ and $2.$ Let $G$ be a planar graph without 4-cycles and 5-cycles. We show that the problem to determine whether $G$ is $(0,k)$-colorable is NP-complete for every positive integer $k.$ Moreover, we construct non-$(1,k)$-colorable planar graphs without 4-cycles and 5-cycles for every positive integer $k.$ In contrast, we prove that $G$ is $(d_1,d_2)$-colorable where $(d_1,d_2)=(4,4),(3,5),$ and $(2,9).$     

Edge choosability of planar graphs without short cycles

In this paper we prove that if G is a planar graph with △= 5 and without 4-cycles or 6-cycles, then G is edge-6-choosable. This consequence together with known results show that, for each fixed k ∈{3,4,5,6}, a k-cycle-free planar graph G is edge-(△ 1)-choosable, where △ denotes the maximum degree of G.     

Group edge choosability of planar graphs without adjacent short cycles

In this paper,we prove that 2-degenerate graphs and some planar graphs without adjacent short cycles are group(Δ(G)+1)-edge-choosable,and some planar graphs with large girth and maximum degree are groupΔ(G)-edge-choosable.     

Adaptable choosability of planar graphs with sparse short cycles

Given a (possibly improper) edge colouring F of a graph G, a vertex colouring of G is adapted toF if no colour appears at the same time on an edge and on its two endpoints. A graph G is called (for some positive integer k) if for any list assignment L to the vertices of G, with |L(v)|≥k for all v, and any edge colouring F of G, G admits a colouring c adapted to F where c(v)∈L(v) for all v. This paper proves that a planar graph G is adaptably 3-choosable if any two triangles in G have distance at least 2 and no triangle is adjacent to a 4-cycle.     

Circular choosability of graphs

This paper discusses the circular version of list coloring of graphs. We give two definitions of the circular list chromatic number (or circular choosability) χ_{c, l}(G) of a graph G and prove that they are equivalent. Then we prove that for any graph G, χ_{c, l}(G) ≥ χ_l(G) ? 1. Examples are given to show that this bound is sharp in the sense that for any ? 0, there is a graph G with χ_{c, l}(G) > χ_l(G) ? 1 + ?. It is also proved that k‐degenerate graphs G have χ_{c, l}(G) ≤ 2k. This bound is also sharp: for each ? < 0, there is a k‐degenerate graph G with χ_{c, l}(G) ≥ 2k ? ?. This shows that χ_{c, l}(G) could be arbitrarily larger than χ_l(G). Finally we prove that if G has maximum degree k, then χ_{c, l}(G) ≤ k + 1. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 210–218, 2005     

Acyclic 5‐choosability of planar graphs without small 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. 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 article, we prove that every planar graph G without 4‐ and 5‐cycles, or without 4‐ and 6‐cycles is acyclically 5‐choosable. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 245–260, 2007     

Equitable list coloring of planar graphs without 4- and 6-cycles

A graph G is equitably k-choosable if for any k-uniform list assignment L, there exists an L-colorable of G such that each color appears on at most vertices. Kostochka, Pelsmajer and West introduced this notion and conjectured that G is equitably k-choosable for k>Δ(G). We prove this for planar graphs with Δ(G)≥6 and no 4- or 6-cycles.     

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