A polynomial algorithm for finding (<Emphasis Type="Italic">g,f</Emphasis>)-colorings orthogonal to stars in bipartite graphs

Let G be a bipartite graph with vertex set V(G) and edge set E(G), and let g and f be two nonnegative integer-valued functions defined on V(G) such that g(x) ≤ f(x) for every vertex x of V(G). A (g, f)-coloring of G is a generalized edge-coloring in which each color appears at each vertex x at least g(x) and at most f(x) times. In this paper a polynomial algorithm to find a (g, f)-coloring of a bipartite graph with some constraints using the minimum number of colors is given. Furthermore, we show that the results in this paper are best possible.     

Vertex coloring complete multipartite graphs from random lists of size 2

Let K_s×m be the complete multipartite graph with s parts and m vertices in each part. Assign to each vertex v of K_s×m a list L(v) of colors, by choosing each list uniformly at random from all 2-subsets of a color set C of size σ(m). In this paper we determine, for all fixed s and growing m, the asymptotic probability of the existence of a proper coloring φ, such that φ(v)∈L(v) for all v∈V(K_s×m). We show that this property exhibits a sharp threshold at σ(m)=2(s−1)m.     

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）．     

图的邻点可区别Ⅵ-全色数和邻点可区别E-全色数

利用穷染、递推的方法讨论了路、圈、完全图、轮和扇的邻点可区别Ⅵ-全染色.并用概率方法研究了一般图的邻点可区别E-全染色,给出了图的邻点可区别E-全色数的一个上界.即δ≥7且△≥28,则有x_(at)~e(G)≤10△,其中δ是图G的最小度,△是图G的最大度.     

Embedding a sequential procedure within an evolutionary algorithm for coloring problems in graphs

We present in this article an evolutionary procedure for solving general optimization problems. The procedure combines efficiently the mechanism of a simple descent method and of genetic algorithms. In order to explore the solution space properly, periods of optimization are interspersed with phases of interaction and diversification. An adaptation of this search principle to coloring problems in graphs is discussed. More precisely, given a graphG, we are interested in finding the largest induced subgraph ofG that can be colored with a fixed numberq of colors. Whenq is larger or equal to the chromatic number ofG, then the problem amounts to finding an ordinary coloring of the vertices ofG.     

三类平方图的点边邻点可区别全色数

根据平方图的结构性质,用穷染,递推的方法,讨论了路,圈,扇的平方图的点边邻点可区别全染色,得到了相应的色数,即并给出了一种染色方案.     

Measurable Sets With Excluded Distances

For a set of distances D = {d ₁,..., d _k } a set A is called D-avoiding if no pair of points of A is at distance d _i for some i. We show that the density of A is exponentially small in k provided the ratios d ₁/d ₂, d ₂/d ₃, …, d _k-1/d _k are all small enough. This resolves a question of Székely, and generalizes a theorem of Furstenberg–Katznelson–Weiss, Falconer–Marstrand, and Bourgain. Several more results on D-avoiding sets are presented. Received: January 2007, Revision: February 2008, Accepted: February 2008     

Mycielski图的BBC染色

基于图G的Mycielski图M（G）,研究xb（G,TG）与xb（M（G）,T’）之间的关系以及xb（G,TG）与xb（M（G）,T＂）之间的关系,其中Tc为G的生成树,T’,T＂分别为M（G）的两类特殊生成树．并给出当G为二部图,完全图以及Halin图时,Xb（M（G）,T＂）的值．     

Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

An interval$t$-coloring of a multigraph $G$ is a proper edge coloring with colors $1,\dots ,t$ such that the colors of the edges incident with every vertex of $G$ are colored by consecutive colors. A cyclic interval$t$-coloring of a multigraph $G$ is a proper edge coloring with colors $1,\dots ,t$ such that the colors of the edges incident with every vertex of $G$ are colored by consecutive colors, under the condition that color $1$ is considered as consecutive to color $t$. Denote by $w\left(G\right)$ ($w_c\left(G\right)$) and $W\left(G\right)$ ($W_c\left(G\right)$) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph $G$, respectively. We present some new sharp bounds on $w\left(G\right)$ and $W\left(G\right)$ for multigraphs $G$ satisfying various conditions. In particular, we show that if $G$ is a $2$-connected multigraph with an interval coloring, then $W\left(G\right)\le 1+?\frac{\left|V\left(G\right)\right|}{2}?(\Delta \left(G\right)?1)$. We also give several results towards the general conjecture that $W_c\left(G\right)\le \left|V\left(G\right)\right|$ for any triangle-free graph $G$ with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most $4$.     

On coloring numbers of graph powers

The weak $r$-coloring numbers ${\mathrm{wcol}}_r\left(G\right)$ of a graph $G$ were introduced by the first two authors as a generalization of the usual coloring number $\mathrm{col}\left(G\right)$, and have since found interesting theoretical and algorithmic applications. This has motivated researchers to establish strong bounds on these parameters for various classes of graphs.Let $G^p$ denote the $p$th power of $G$. We show that, all integers $p>0$ and $\Delta \ge 3$ and graphs $G$ with $\Delta \left(G\right)\le \Delta$ satisfy $\mathrm{col}\left(G^p\right)\in O\left(p\cdot {\mathrm{wcol}}_{\lceil p∕2\rceil }\left(G\right){(\Delta -1)}^{\lfloor p∕2\rfloor }\right)$; for fixed tree width or fixed genus the ratio between this upper bound and worst case lower bounds is polynomial in $p$. For the square of graphs $G$, we also show that, if the maximum average degree $2k-2<\mathrm{mad}\left(G\right)\le 2k$, then $\mathrm{col}\left(G^2\right)\le (2k-1)\Delta \left(G\right)+2k+1$.