Edge‐partitions of planar graphs and their game coloring numbers

Let G be a planar graph 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 (i) Δ(H) ≤ 4 if g(G) ≥ 5; (ii) Δ(H) ≤ 2 if g(G) ≥ 7; (iii) Δ(H)≤ 1 if g(G) ≥ 11; (iv) Δ(H) ≤ 7 if G does not contain 4‐cycles (though it may contain 3‐cycles). These results are applied to find the following upper bounds for the game coloring number col_g(G) of a planar graph G: (i) col_g(G) ≤ 8 if g(G) ≥ 5; (ii) col_g(G)≤ 6 if g(G) ≥ 7; (iii) col_g(G) ≤ 5 if g(G) ≥ 11; (iv) col_g(G) ≤ 11 if G does not contain 4‐cycles (though it may contain 3‐cycles). © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 307–317, 2002     

On a conjecture by Plummer and Toft

The cyclic chromatic number χ_c(G) of a 2‐connected plane graph G is the minimum number of colors in an assigment of colors to the vertices of G such that, for every face‐bounding cycle f of G, the vertices of f have different colors. Plummer and Toft proved that, for a 3‐connected plane graph G, under the assumption Δ*(G) ≥ 42, where Δ*(G) is the size of a largest face of G, it holds that χ_c(G) ≤ Δ*(G) + 4. They conjectured that, if G is a 3‐connected plane graph, then χ_c>(G) ≤ Δ*(G) + 2. In the article the conjecture is proved for Δ*(G) ≥ 24. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 177–189, 1999     

Improper choosability of graphs and maximum average degree

Improper choosability of planar graphs has been widely studied. In particular, ?krekovski investigated the smallest integer g_k such that every planar graph of girth at least g_k is k‐improper 2‐choosable. He proved [9] that 6 ≤ g₁ ≤ 9; 5 ≤ g₂ ≤ 7; 5 ≤ g₃ ≤ 6; and ? k ≥ 4, g_k = 5. In this article, we study the greatest real M(k, l) such that every graph of maximum average degree less than M(k, l) is k‐improper l‐choosable. We prove that if l ≥ 2 then . As a corollary, we deduce that g₁ ≤ 8 and g₂ ≤ 6, and we obtain new results for graphs of higher genus. We also provide an upper bound for M(k, l). This implies that for any fixed l, . © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 181–199, 2006     

New upper bounds on linear coloring of planar graphs

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, it is proved that every planar graph G with girth g and maximum degree Δ has(1)lc(G) ≤Δ 21 if Δ≥ 9; (2)lc(G) ≤「Δ/2」 + 7 ifg ≥ 5; (3) lc(G) ≤「Δ/2」 + 2 ifg ≥ 7 and Δ≥ 7.     

Coloring the square of a planar graph

We prove that for any planar graph G with maximum degree Δ, it holds that the chromatic number of the square of G satisfies χ(G²) ≤ 2Δ + 25. We generalize this result to integer labelings of planar graphs involving constraints on distances one and two in the graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 110–124, 2003     

Acyclic edge colorings of graphs

A proper coloring of the edges 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. For certain graphs G, a′(G) ≥ Δ(G) + 2 where Δ(G) is the maximum degree in G. It is known that a′(G) ≤ 16 Δ(G) for any graph G. We prove that there exists a constant c such that a′(G) ≤ Δ(G) + 2 for any graph G whose girth is at least cΔ(G) log Δ(G), and conjecture that this upper bound for a′(G) holds for all graphs G. We also show that a′(G) ≤ Δ + 2 for almost all Δ‐regular graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 157–167, 2001     

New upper bounds for the chromatic number of a graph*

We show that for any graph G, the chromatic number χ(G) ≤ Δ₂(G) + 1, where Δ₂(G) is the largest degree that a vertex ν can have subject to the condition that ν is adjacent to a vertex whose degree is at least as big as its own. Moreover, we show that the upper bound is best possible in the the following sense: If Δ₂(G) ≥ 3, then to determine whether χ(G) ≤ Δ₂(G) is an NP‐complete problem. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 117–120, 2001     

Edge‐face chromatic number and edge chromatic number of simple plane graphs

Given a simple plane graph G, an edge‐face k‐coloring of G is a function ? : E(G) ∪ F(G) → {1,…,k} such that, for any two adjacent or incident elements a, b ∈ E(G) ∪ F(G), ?(a) ≠ ?(b). Let χ_e(G), χ_ef(G), and Δ(G) denote the edge chromatic number, the edge‐face chromatic number, and the maximum degree of G, respectively. In this paper, we prove that χ_ef(G) = χ_e(G) = Δ(G) for any 2‐connected simple plane graph G with Δ (G) ≥ 24. © 2005 Wiley Periodicals, Inc. J Graph Theory     

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     

A new upper bound on the cyclic chromatic number

A cyclic coloring of a plane graph is a vertex coloring such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a graph is its cyclic chromatic number χ^c. Let Δ^* be the maximum face degree of a graph. There exist plane graphs with χ^c = ?3/2 Δ^*?. Ore and Plummer [ 5 ] proved that χ^c ≤ 2, Δ^*, which bound was improved to ?9/5, Δ^*? by Borodin, Sanders, and Zhao [ 1 ], and to ?5/3,Δ^*? by Sanders and Zhao [ 7 ]. We introduce a new parameter k^*, which is the maximum number of vertices that two faces of a graph can have in common, and prove that χ^c ≤ max {Δ^* + 3,k^* + 2, Δ^* + 14, 3, k^* + 6, 18}, and if Δ^* ≥ 4 and k^* ≥ 4, then χ^c ≤ Δ^* + 3,k^* + 2. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Acyclic chromatic indices of planar graphs with large girth

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a^′(G) of G is the smallest k such that G has an acyclic edge coloring using k colors.In this paper, we prove that every planar graph G with girth g(G) and maximum degree Δ has a^′(G)=Δ if there exists a pair (k,m)∈{(3,11),(4,8),(5,7),(8,6)} such that G satisfies Δ≥k and g(G)≥m.     

Equitable coloring of random graphs

An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most one. The least positive integer k for which there exists an equitable coloring of a graph G with k colors is said to be the equitable chromatic number of G and is denoted by χ₌(G). The least positive integer k such that for any k′ ≥ k there exists an equitable coloring of a graph G with k′ colors is said to be the equitable chromatic threshold of G and is denoted by χ₌*(G). In this paper, we investigate the asymptotic behavior of these coloring parameters in the probability space G(n,p) of random graphs. We prove that if n^?1/5+? < p < 0.99 for some 0 < ?, then almost surely χ(G(n,p)) ≤ χ₌(G(n,p)) = (1 + o(1))χ(G(n,p)) holds (where χ(G(n,p)) is the ordinary chromatic number of G(n,p)). We also show that there exists a constant C such that if C/n < p < 0.99, then almost surely χ(G(n,p)) ≤ χ₌(G(n,p)) ≤ (2 + o(1))χ(G(n,p)). Concerning the equitable chromatic threshold, we prove that if n^?(1??) < p < 0.99 for some 0 < ?, then almost surely χ(G(n,p)) ≤ χ₌* (G(n,p)) ≤ (2 + o(1))χ(G(n,p)) holds, and if < p < 0.99 for some 0 < ?, then almost surely we have χ(G(n,p)) ≤ χ₌*(G(n,p)) = O_?(χ(G(n,p))). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Linear coloring of graphs embeddable in a surface of nonnegative characteristic

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, we prove that every graph G with girth g(G) and maximum degree Δ(G) that can be embedded in a surface of nonnegative characteristic has lc(G) = Δ(2G )+ 1 if there is a pair (Δ, g) ∈ {(13, 7), (9, 8), (7, 9), (5, 10), (3, 13)} such that G s...     

Coloring graphs from lists with bounded size of their union

A graph G is k‐choosable if its vertices can be colored from any lists L(ν) of colors with |L(ν)| ≥ k for all ν ∈ V(G). A graph G is said to be (k,?)‐choosable if its vertices can be colored from any lists L(ν) with |L(ν)| ≥k, for all ν∈ V(G), and with . For each 3 ≤ k ≤ ?, we construct a graph G that is (k,?)‐choosable but not (k,? + 1)‐choosable. On the other hand, it is proven that each (k,2k ? 1)‐choosable graph G is O(k · ln k · 2^4k)‐choosable. © 2005 Wiley Periodicals, Inc. J Graph Theory     

On total 9‐coloring planar graphs of maximum degree seven

Given a graph G, a total k‐coloring of G is a simultaneous coloring of the vertices and edges of G with at most k colors. If Δ(G) is the maximum degree of G, then no graph has a total Δ‐coloring, but Vizing conjectured that every graph has a total (Δ + 2)‐coloring. This Total Coloring Conjecture remains open even for planar graphs. This article proves one of the two remaining planar cases, showing that every planar (and projective) graph with Δ ≤ 7 has a total 9‐coloring by means of the discharging method. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 67–73, 1999     

Isometric cycles,cutsets, and crowning of bridged graphs

A graph G is bridged if every cycle C of length at least 4 has vertices x,y such that d_G(x,y) < d_C(x,y). A cycle C is isometric if d_G(x,y) = d_C(x,y) for all x,y ∈ V(C). We show that every graph contractible to a graph with girth g has an isometric cycle of length at least g. We use this to show that every minimal cutset S in a bridged graph G induces a connected subgraph. We introduce a “crowning” construction to enlarge bridged graphs. We use this to construct examples showing that for every connected simple graph H with girth at least 6 (including trees), there exists a bridged graph G such that G has a unique minimum cutset S and that G[S] = H. This provides counterexamples to Hahn's conjecture that d_G(u,v) ≤ 2 when u and v lie in a minimum cutset in a bridged graph G. We also study the convexity of cutsets in bridged graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 161–170, 2003     

Acyclic edge colorings of planar graphs and seriesparallel graphs

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. For planar graphs G with girth g(G), we prove that a′(G) ⩽ max{2Δ(G) − 2, Δ(G) + 22} if g(G) ⩾ 3, a′(G) ⩽ Δ(G) + 2 if g(G) ⩾ 5, a′(G) ⩽ Δ(G) + 1 if g(G) ⩾ 7, and a′(G) = Δ(G) if g(G) ⩾ 16 and Δ(G) ⩾ 3. For series-parallel graphs G, we have a′(G) ⩽ Δ(G) + 1. This work was supported by National Natural Science Foundation of China (Grant No. 10871119) and Natural Science Foundation of Shandong Province (Grant No. Y2008A20).     

The circular chromatic number of series‐parallel graphs with large girth

It was proved by Hell and Zhu that, if G is a series‐parallel graph of girth at least 2⌊(3k − 1)/2⌋, then χ_c(G) ≤ 4k/(2k − 1). In this article, we prove that the girth requirement is sharp, i.e., for any k ≥ 2, there is a series‐parallel graph G of girth 2⌊(3k − 1)/2⌋ − 1 such that χ_c(G) > 4k/(2k − 1). © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 185–198, 2000     

A note on list edge and list total coloring of planar graphs without adjacent short cycles

LetGbe a planar graph with maximum degreeΔ.In this paper,we prove that if any4-cycle is not adjacent to ani-cycle for anyi∈{3,4}in G,then the list edge chromatic numberχl(G)=Δand the list total chromatic numberχl(G)=Δ+1.