Planar graphs without 5-cycles or without 6-cycles

Let G be a planar graph without 5-cycles or without 6-cycles. In this paper, we prove that if G is connected and δ(G)≥2, then there exists an edge xy∈E(G) such that d(x)+d(y)≤9, or there is a 2-alternating cycle. By using the above result, we obtain that (1) its linear 2-arboricity , (2) its list total chromatic number is Δ(G)+1 if Δ(G)≥8, and (3) its list edge chromatic number is Δ(G) if Δ(G)≥8.     

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].     

Total colorings of planar graphs without adjacent triangles

Let G be a planar graph without adjacent 3-cycles, that is, two cycles of length 3 are not incident with a common edge. In this paper, it is proved that the total coloring conjecture is true for G; moreover, if Δ(G)≥9, then the total chromatic number χ^″(G) of G is Δ(G)+1. Some other related results are obtained, too.     

Planar graphs without specific cycles are 2-degenerate

A graph G is k-degenerate if each subgraph of G has a vertex of degree at most k. It is known that every simple planar graph with girth 6, or equivalently without 3-, 4-, and 5-cycles, is 2-degenerate. In this work, we investigate for which k every planar graph without 4-, 6-, … , and 2k-cycles is 2-degenerate. We determine that k is 5 and the result is tight since the truncated dodecahedral graph is a 3-regular planar graph without 4-, 6-, and 8-cycles. As a related result, we also show that every planar graph without 4-, 6-, 9-, and 10-cycles is 2-degenerate.     

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.     

Planar graphs with maximum degree 8 and without intersecting chordal 4-cycles are 9-totally colorable

The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ’’ (G). It is shown 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 intersecting chordal 4-cycles, then χ ’’(G) = 9.     

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.     

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.     

Edge-partitions of graphs of nonnegative characteristic and their game coloring numbers

Let G be a graph of nonnegative characteristic and let g(G) and Δ(G) be its girth and maximum degree, respectively. We show that G has an edge-partition into a forest and a subgraph H so that (1) Δ(H)?1 if g(G)?11; (2) Δ(H)?2 if g(G)?7; (3) Δ(H)?4 if either g(G)?5 or G does not contain 4-cycles and 5-cycles; (4) Δ(H)?6 if G does not contain 4-cycles. These results are applied to find the following upper bounds for the game coloring number col_g(G) of G: (1) col_g(G)?5 if g(G)?11; (2) col_g(G)?6 if g(G)?7; (3) col_g(G)?8 if either g(G)?5 or G contains no 4-cycles and 5-cycles; (4) col_g(G)?10 if G does not contain 4-cycles.     

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.     

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.     

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.     

Matching Properties in Total Domination Vertex Critical Graphs

A vertex subset S of a graph G = (V,E) is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number of G, denoted by γ _t (G), is the minimum cardinality of a total dominating set of G. A graph G with no isolated vertex is said to be total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γ _t (G?v) < γ _t (G). A total domination vertex critical graph G is called k-γ _t -critical if γ _t (G) = k. In this paper we first show that every 3-γ _t -critical graph G of even order has a perfect matching if it is K _1,5-free. Secondly, we show that every 3-γ _t -critical graph G of odd order is factor-critical if it is K _1,5-free. Finally, we show that G has a perfect matching if G is a K _1,4-free 4-γ _t (G)-critical graph of even order and G is factor-critical if G is a K _1,4-free 4-γ _t (G)-critical graph of odd order.     

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.     

The Linear Arboricity of Planar Graphs without 5-, 6-Cycles with Chords

The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. In this paper, it is proved that for a planar graph G, ${la(G)=\lceil\frac{\Delta(G)}{2}\rceil}$ if Δ(G) ≥ 7 and G has no 5-cycles with chords, or Δ(G) ≥ 5 and G has no 5-, 6-cycles with chords.     

The Size of Maximally Irregular Graphs and Maximally Irregular Triangle-Free Graphs

Let G be a graph. The irregularity index of G, denoted by t(G), is the number of distinct values in the degree sequence of G. For any graph G, t(G) ≤ Δ(G), where Δ(G) is the maximum degree. If t(G) = Δ(G), then G is called maximally irregular. In this paper, we give a tight upper bound on the size of maximally irregular graphs, and prove the conjecture proposed in [6] on the size of maximally irregular triangle-free graphs. Extremal graphs are also characterized.     

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].     

On equality in an upper bound for the restrained and total domination numbers of a graph

Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set (RDS) if every vertex not in S is adjacent to a vertex in S and to a vertex in V?S. The restrained domination number of G, denoted by γ_r(G), is the minimum cardinality of an RDS of G. A set S⊆V is a total dominating set (TDS) if every vertex in V is adjacent to a vertex in S. The total domination number of a graph G without isolated vertices, denoted by γ_t(G), is the minimum cardinality of a TDS of G.Let δ and Δ denote the minimum and maximum degrees, respectively, in G. If G is a graph of order n with δ?2, then it is shown that γ_r(G)?n-Δ, and we characterize the connected graphs with δ?2 achieving this bound that have no 3-cycle as well as those connected graphs with δ?2 that have neither a 3-cycle nor a 5-cycle. Cockayne et al. [Total domination in graphs, Networks 10 (1980) 211-219] showed that if G is a connected graph of order n?3 and Δ?n-2, then γ_t(G)?n-Δ. We further characterize the connected graphs G of order n?3 with Δ?n-2 that have no 3-cycle and achieve γ_t(G)=n-Δ.     

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.     

Toughness and the existence of fractional k-factors of graphs

The toughness of a graph G, t(G), is defined as t(G)=min{|S|/ω(G-S)|S⊆V(G),ω(G-S)>1} where ω(G-S) denotes the number of components of G-S or t(G)=+∞ if G is a complete graph. Much work has been contributed to the relations between toughness and the existence of factors of a graph. In this paper, we consider the relationship between the toughness and the existence of fractional k-factors. It is proved that a graph G has a fractional 1-factor if t(G)?1 and has a fractional k-factor if t(G)?k-1/k where k?2. Furthermore, we show that both results are best possible in some sense.