An oriented coloring of planar graphs with girth at least five

An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented graph with a maximum average degree less than and girth at least 5 has an oriented chromatic number at most 16. This implies that every oriented planar graph with girth at least 5 has an oriented chromatic number at most 16, that improves the previous known bound of 19 due to Borodin et al. [O.V. Borodin, A.V. Kostochka, J. Nešet?il, A. Raspaud, É. Sopena, On the maximum average degree and the oriented chromatic number of a graph, Discrete Math. 206 (1999) 77-89].     

Dense graphs with small clique number

We consider the structure of K_r‐free graphs with large minimum degree, and show that such graphs with minimum degree δ>(2r ? 5)n/(2r ? 3) are homomorphic to the join K_{r ? 3}∨H, where H is a triangle‐free graph. In particular this allows us to generalize results from triangle‐free graphs and show that K_r‐free graphs with such a minimum degree have chromatic number at most r +1. We also consider the minimum‐degree thresholds for related properties. Copyright © 2010 John Wiley & Sons, Ltd. 66:319‐331, 2011     

Upper bounds on the linear chromatic number of a graph

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is a union of vertex-disjoint paths. The linear chromatic number of G is the smallest number of colors in a linear coloring of G.Let G be a graph with maximum degree Δ(G). In this paper we prove the following results: (1) ; (2) if Δ(G)≤4; (3) if Δ(G)≤5; (4) if G is planar and Δ(G)≥52.     

On the oriented chromatic number of graphs with given excess

The excess of a graph G is defined as the minimum number of edges that must be deleted from G in order to get a forest. We prove that every graph with excess at most k has chromatic number at most and that this bound is tight. Moreover, we prove that the oriented chromatic number of any graph with excess k is at most k+3, except for graphs having excess 1 and containing a directed cycle on 5 vertices which have oriented chromatic number 5. This bound is tight for k?4.     

平面图线性色数的一个上界

如果图G的一个正常染色满足染任意两种颜色的顶点集合导出的子图是一些点不交的路的并,则称这个正常染色为图G的线性染色．图G的线性色数用lc（G）表示,是指G的所有线性染色中所用的最少颜色的个数．证明了：若G是一个最大度△（G）≠5,6的平面图,则lc（G）≤2△（G）．     

Clique coloring of dense random graphs

The clique chromatic number of a graph is the minimum number of colors in a vertex coloring so that no maximal (with respect to containment) clique is monochromatic. We prove that the clique chromatic number of the binomial random graph is, with high probability, . This settles a problem of McDiarmid, Mitsche, and Prałat who proved that it is with high probability.     

Acyclick-strong coloring of maps on surfaces

A coloring of graph vertices is called acyclic if the ends of each edge are colored in distinct colors and there are no two-colored cycles. Suppose each face of rank not greater thank, k ≥ 4, on a surfaceS ^N is replaced by the clique on the same set of vertices. Then the pseudograph obtained in this way can be colored acyclically in a set of colors whose cardinality depends linearly onN and onk. Results of this kind were known before only for 1 ≤N ≤ 2 and 3 ≤k ≤ 4. Translated fromMatematicheskie Zametki, vol. 67, No. 1, pp. 36–45, January, 2000.     

Injective colorings of sparse graphs

Let denote the maximum average degree (over all subgraphs) of G and let χ_i(G) denote the injective chromatic number of G. We prove that if , then χ_i(G)≤Δ(G)+1; and if , then χ_i(G)=Δ(G). Suppose that G is a planar graph with girth g(G) and Δ(G)≥4. We prove that if g(G)≥9, then χ_i(G)≤Δ(G)+1; similarly, if g(G)≥13, then χ_i(G)=Δ(G).     

Linear coloring of planar graphs with large girth

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 of the graph G is the smallest number of colors in a linear coloring of G. In this paper we prove that every planar graph G with girth g and maximum degree Δ has if G satisfies one of the following four conditions: (1) g≥13 and Δ≥3; (2) g≥11 and Δ≥5; (3) g≥9 and Δ≥7; (4) g≥7 and Δ≥13. Moreover, we give better upper bounds of linear chromatic number for planar graphs with girth 5 or 6.     

On circle graphs with girth at least five

Circle graphs with girth at least five are known to be 2-degenerate [A.A. Ageev, Every circle graph with girth at least 5 is 3-colourable, Discrete Math. 195 (1999) 229-233]. In this paper, we prove that circle graphs with girth at least g≥5 and minimum degree at least two contain a chain of g−4 vertices of degree two, which implies Ageev’s result in the case g=5. We then use this structural property to give an upper bound on the circular chromatic number of circle graphs with girth at least g≥5 as well as a precise estimate of their maximum average degree.     

An improved upper bound on the adjacent vertex distinguishing total chromatic number of graphs

An adjacent vertex distinguishing total $k$-coloring of a graph $G$ is a proper total $k$-coloring of $G$ such that any pair of adjacent vertices have different sets of colors. The minimum number $k$ needed for such a total coloring of $G$ is denoted by ${\chi }_a^{\prime \prime }\left(G\right)$. In this paper we prove that ${\chi }_a^{\prime \prime }\left(G\right)\le 2\Delta \left(G\right)?1$ if $\Delta \left(G\right)\ge 4$, and ${\chi }_a^{\prime \prime }\left(G\right)\le ?\frac{5\Delta \left(G\right)+8}{3}?$ in general. This improves a result in Huang et al. (2012) which states that ${\chi }_a^{\prime \prime }\left(G\right)\le 2\Delta \left(G\right)$ for any graph with $\Delta \left(G\right)\ge 3$.     

Planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable

The Total Coloring Conjecture, in short, TCC, says that every simple graph is (Δ+2)-totally-colorable where Δ is the maximum degree of the graph. Even for planar graphs this conjecture has not been completely settled yet. However, every planar graph with Δ≥9 has been proved to be (Δ+1)-totally-colorable. In this paper, we prove that planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable.     

Smaller planar triangle-free graphs that are not 3-list-colorable

In 1995, Voigt constructed a planar triangle-free graph that is not 3-list-colorable. It has 166 vertices. Gutner then constructed such a graph with 164 vertices. We present two more graphs with these properties. The first graph has 97 vertices and a failing list assignment using triples from a set of six colors, while the second has 109 vertices and a failing list assignment using triples from a set of five colors.     

围长至少为5的平面图的线性染色

如果图G的一个正常染色满足染任意两种颜色的顶点集合导出的子图是一些点不交的路的并,则称这个正常染色为图G的线性染色.图G的线性色数用lc(G)表示,是指G的所有线性染色中所用的最少颜色的个数本文证明了对于每一个最大度为△(G)且围长至少为5的平面图G有lc(G)≤[△(G)/2]+5,并且当△(G)∈{7,8,…,14...     

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

An Erratum has been published for this article in Journal of Graph Theory 48: 329–330, 2005 . Let M be a set of positive integers. The distance graph generated by M, denoted by G(Z, M), has the set Z of all integers as the vertex set, and edges ij whenever |i?j| ∈ M. We investigate the fractional chromatic number and the circular chromatic number for distance graphs, and discuss their close connections with some number theory problems. In particular, we determine the fractional chromatic number and the circular chromatic number for all distance graphs G(Z, M) with clique size at least |M|, except for one case of such graphs. For the exceptional case, a lower bound for the fractional chromatic number and an upper bound for the circular chromatic number are presented; these bounds are sharp enough to determine the chromatic number for such graphs. Our results confirm a conjecture of Rabinowitz and Proulx 22 on the density of integral sets with missing differences, and generalize some known results on the circular chromatic number of distance graphs and the parameter involved in the Wills' conjecture 26 (also known as the “lonely runner conjecture” 1 ). © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 129–146, 2004     

List r-hued chromatic number of graphs with bounded maximum average degrees

For integers $k,r>0$, a $(k,r)$-coloring of a graph $G$ is a proper coloring $c$ with at most $k$ colors such that for any vertex $v$ with degree $d\left(v\right)$, there are at least min$\{d\left(v\right),r\}$ different colors present at the neighborhood of $v$. The $r$-hued chromatic number of $G$, ${\chi }_r\left(G\right)$, is the least integer $k$ such that a $(k,r)$-coloring of $G$ exists. The list$r$-hued chromatic number${\chi }_{L,r}\left(G\right)$ of $G$ is similarly defined. Thus if $\Delta \left(G\right)\ge r$, then ${\chi }_{L,r}\left(G\right)\ge {\chi }_r\left(G\right)\ge r+1$. We present examples to show that, for any sufficiently large integer $r$, there exist graphs with maximum average degree less than 3 that cannot be $(r+1,r)$-colored. We prove that, for any fraction $q<\frac{14}{5}$, there exists an integer $R=R\left(q\right)$ such that for each $r\ge R$, every graph $G$ with maximum average degree $q$ is list $(r+1,r)$-colorable. We present examples to show that for some $r$ there exist graphs with maximum average degree less than 4 that cannot be $r$-hued colored with less than $\frac{3r}{2}$ colors. We prove that, for any sufficiently small real number $?>0$, there exists an integer $h=h\left(?\right)$ such that every graph $G$ with maximum average degree $4??$ satisfies ${\chi }_{L,r}\left(G\right)\le r+h\left(?\right)$. These results extend former results in Bonamy et al. (2014).