Planar graphs without mutually adjacent 3-, 5-, and 6-cycles are 3-degenerate

A graph G is k-degenerate if every subgraph of G has a vertex with degree at most k. Using the Euler's formula, one can obtain that planar graphs without 3-cycles are 3-degenerate. Wang and Lih, and Fijav? et al. proved the analogue results for planar graphs without 5-cycles and planar graphs without 6-cycles, respectively. Recently, Liu et al. showed that planar graphs without 3-cycles adjacent to 5-cycles are 3-degenerate. In this work, we generalized all aforementioned results by showing that planar graphs without mutually adjacent 3-,5-, and 6-cycles are 3-degenerate. A graph G without mutually adjacent 3-,5-, and 6-cycles means that G cannot contain three graphs, say $G_1,G_2$, and $G_3$, where $G_1$ is a 3-cycle, $G_2$ is a 5-cycle, and $G_3$ is a 6-cycle such that each pair of $G_1,G_2$, and $G_3$ are adjacent. As an immediate consequence, we have that every planar graph without mutually adjacent 3-,5-, and 6-cycles is DP-4-colorable. This consequence also generalizes the result by Chen et al that planar graphs without 5-cycles adjacent to 6-cycles are DP-4-colorable.     

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.     

Bordeaux 3-color conjecture and 3-choosability

A graph G=(V,E) is list L-colorable if for a given list assignment L={L(v):v∈V}, there exists a proper coloring c of G such that c(v)∈L(v) for all v∈V. If G is list L-colorable for every list assignment with |L(v)|?k for all v∈V, then G is said to be k-choosable.In this paper, we prove that (1) every planar graph either without 4- and 5-cycles, and without triangles at distance less than 4, or without 4-, 5- and 6-cycles, and without triangles at distance less than 3 is 3-choosable; (2) there exists a non-3-choosable planar graph without 4-cycles, 5-cycles, and intersecting triangles. These results have some consequences on the Bordeaux 3-color conjecture by Borodin and Raspaud [A sufficient condition for planar graphs to be 3-colorable. J. Combin. Theory Ser. B 88 (2003) 17-27].     

Structural properties and edge choosability of planar graphs without 4-cycles

Some structural properties of planar graphs without 4-cycles are investigated. By the structural properties, it is proved that every planar graph G without 4-cycles is edge-(Δ(G)+1)-choosable, which perfects the result given by Zhang and Wu: If G is a planar graph without 4-cycles, then G is edge-t-choosable, where t=7 if Δ(G)=5, and otherwise t=Δ(G)+1.     

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.     

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].     

A note on list improper coloring planar graphs

A graph G is called (k, d)^*-choosable if, for every list assignment L satisfying |L(v)| = k for all v ϵ V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. In this note, we prove that every planar graph without 4-cycles and l-cycles for some l ϵ {5, 6, 7} is (3, 1)^*-choosable.     

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 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.     

A sufficient condition for planar graphs with maximum degree 8 to be 9-totally colorable

A total k-coloring of a graph G is a coloring of V(G) ∪ E(G) using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number χ'(G) is the smallest integer k such that G has a total k-coloring. It is known that if a planar graph G has maximum degree Δ≥ 9, then χ'(G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without a fan of four adjacent 3-cycles, then χ'(G) = 9.     

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.     

On total chromatic number of planar graphs without 4-cycles

Let G be a simple graph with maximum degree A(G) and total chromatic number Xve(G). Vizing conjectured thatΔ(G) 1≤Xve(G)≤Δ(G) 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has not been settled yet. The unsettled difficult case for planar graphs isΔ(G) = 6. This paper shows that if G is a simple planar graph with maximum degree 6 and without 4-cycles, then Xve(G)≤8. Together with the previous results on this topic, this shows that every simple planar graph without 4-cycles satisfies the Total Chromatic Conjecture.     

Edge-choosability of planar graphs without adjacent triangles or without 7-cycles

A graph G is edge-L-colorable, if for a given edge assignment L={L(e):e∈E(G)}, there exists a proper edge-coloring ? of G such that ?(e)∈L(e) for all e∈E(G). If G is edge-L-colorable for every edge assignment L with |L(e)|≥k for e∈E(G), then G is said to be edge-k-choosable. In this paper, we prove that if G is a planar graph with maximum degree Δ(G)≠5 and without adjacent 3-cycles, or with maximum degree Δ(G)≠5,6 and without 7-cycles, then G is edge-(Δ(G)+1)-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).     

Fault tolerance of vertex pancyclicity in alternating group graphs

We study fault tolerance of vertex k pancyclicity of the alternating group graph AG_n. A graph G is vertex k pancyclic, if for every vertex p∈G, there is a cycle going through p of every length from k to |G|. Xue and Liu [Z.-J. Xue, S.-Y. Liu, An optimal result on fault-tolerant cycle-embedding in alternating group graphs, Inform. Proc. Lett. 109 (2009) 1197-1201] put the conjecture that AG_n is (2n-7)-fault-tolerant vertex pancyclic, which means that if the number of faults |F|?2n-7, then AG_n-F is still vertex pancyclic. Chang and Yang [J.-M. Chang, J.-S. Yang, Fault-tolerant cycle-embedding in alternating group graphs, Appl. Math. Comput. 197 (2008) 760-767] showed that for the shortest cycles the fault-tolerance of AG_n is much lower. They noted that with n-2 faults one can cut all 3-cycles going through a given vertex p (it is easy to observe that the same set of faults cuts all 4- and 5-cycles going through p). On the other hand they show that AG_n is n-3-fault tolerant vertex 3 pancyclic. In this paper we show that the cycles of length ?6 are much more fault-tolerant. More precisely, we show that AG_n is (2n-6)-fault-tolerant vertex 6 pancyclic. This bound is optimal, because every vertex p has 2n-4 neighbors.     

Sequences realizable by maximal k-degenerate graphs

A graph G is called k-degenerate if every subgraph of G has a vertex of degree at most k. A k-degenerate graph G is maximal k-degenerate if for every edge e ? E(G), G + e is not k-degenerate. Necessary and sufficient conditions for the sequence II = (d₁, d₂, ?, d_p) to be a degree sequence of a maximal k-degenerate graph G are presented. © 1995 John Wiley & Sons, Inc.     

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.     

Every signed planar graph without cycles of length from 4 to 8 is 3-colorable

In this paper, we investigate the signed graph version of Erdös problem: Is there a constant $c$ such that every signed planar graph without $k$-cycles, where $4\le k\le c$, is 3-colorable and prove that each signed planar graph without cycles of length from 4 to 8 is 3-colorable.     

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.