由轮图导出的某些平面图类的星色数

图的星色数是通常色数概念的推广.本文求出了几类由轮图导出的平面图的星色数.前两类是由3-或5-轮图经细分等构造出的,其星色数分别为2+2/(2n+1),2+3/(3n+1)和2+3/(3n-1).第三类平面图是由n-轮图经过Hajos构造得到的,其星色数为3+1/n.本类图的星色数结果推广了已有结论.     

Star chromatic numbers of some planar graphs

The concept of the star chromatic number of a graph was introduced by Vince (A. Vince, Star chromatic number, J. Graph Theory 12 (1988), 551–559), which is a natural generalization of the chromatic number of a graph. This paper calculates the star chromatic numbers of three infinite families of planar graphs. More precisely, the first family of planar graphs has star chromatic numbers consisting of two alternating infinite decreasing sequences between 3 and 4; the second family of planar graphs has star chromatic numbers forming an infinite decreasing sequence between 3 and 4; and the third family of planar graphs has star chromatic number 7/2. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 33–42, 1998     

图的星色数

给出了一些星色数为４的平面图,它们不含有轮图作为子图,这回答了Ｚｈｕ的一个问题,给出了一类４连通平面图其星色数在３与４之间,这也回答了Ａｂｂｏｔｔ和Ｚｈｏｕ的一个问题,应用图的同态概念,讨论了某些图的字典积的星色数,证明了一个图及其补图的星色数的和与积所满足的两个不等式。     

一类星色数为4的平面图

图的星色数的概念是Vince在1988年提出的,它是图的色数的一个推广.本文构造了一类星色数是4的平面图.     

Randomized online graph coloring

We study the problem of coloring graphs in an online manner. The only known deterministic online graph coloring algorithm with a sublinear performance function was found by [9.], 319–325). Their algorithm colors graphs of chromatic number χ with no more than (2χn)/log^* n colors, where n is the number of vertices. They point out that the performance can be improved slightly for graphs with bounded chromatic number. For three-chromatic graphs the number of colors used, for example, is O(n log log log n/log log n). We show that randomization helps in coloring graphs online. We present a simple randomized online algorithm to color graphs with expected number of colors O(2^χχ²n^{(χ−2)/(χ−1)}(log n)^1/(χ−1)). For three-colorable graphs the expected number of colors our algorithm uses is . All our algorithms run in polynomial time. It is interesting to note that our algorithm compares well with the best known polynomial time offline algorithms. For instance, the best polynomial time algorithm known for three-colorable graphs, due to [4.] pp. 554–562). We also prove a lower bound of Ω((1/(χ − 1))((log n/(12(χ + 1))) − 1)^χ−1) for the randomized model. No lower bound for the randomized model was previously known. For bounded χ, our result improves even the best known lower bound for the deterministic case: Ω((log n/log log n)^χ−1), due to Noga Alon (personal communication, September 1989).     

推广的奇轮的圆色数

图G的圆色数(又称"星色数")xc(G)是Vince在1988年提出的,它是图的色数 的自然推广.本文由奇轮出发构造了一族平面图,并证明了此类图的圆色数恰恰介于2和 3之间,填补了该领域的空白.     

Proof of a conjecture on fractional Ramsey numbers

Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function r_f (a₁, a₂, …, a_k) as an extension of the classical definition for Ramsey numbers. They determined an exact formula for the fractional Ramsey function for the case k=2. In this article, we answer an open problem by determining an explicit formula for the general case k>2 by constructing an infinite family of circulant graphs for which the independence numbers can be computed explicitly. This construction gives us two further results: a new (infinite) family of star extremal graphs which are a superset of many of the families currently known in the literature, and a broad generalization of known results on the chromatic number of integer distance graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 164–178, 2010     

The fractional chromatic number of mycielski's graphs

James Propp The most familiar construction of graphs whose clique number is much smaller than their chromatic number is due to Mycielski, who constructed a sequence G_n of triangle-free graphs with X(G_n) = n. In this article, we calculate the fractional chromatic number of G_n and show that this sequence of numbers satisfies the unexpected recurrence a_n+1 = a_n + (1/a_n). © 1995 John Wiley & Sons, Inc.     

On the chromatic number of random graphs

We consider random graphs G_n,p with fixed edge-probability p. We refine an argument of Bollobás to show that almost all such graphs have chromatic number equal to n/{2 log_b n ? 2 log_b log_b n + O(1)} where b = 1/(1 ? p).     

The star-chromatic number of planar graphs

The star-chromatic number of a graph, a parameter introduced by Vince, is a natural generalization of the chromatic number of a graph. Here we construct planar graphs with star-chromatic number r, where r is any rational number between 2 and 3, partially answering a question of Vince. © 1997 John Wiley & Sons, Inc.     

Game coloring the Cartesian product of graphs

This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G^* be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G^* has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product G□G′ has game chromatic number at most k(k + m ? 1). As a consequence, the Cartesian product of two forests has game chromatic number at most 10, and the Cartesian product of two planar graphs has game chromatic number at most 105. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 261–278, 2008     

Star chromatic numbers of graphs

We investigate the relation between the star-chromatic number (G) and the chromatic number (G) of a graphG. First we give a sufficient condition for graphs under which their starchromatic numbers are equal to their ordinary chromatic numbers. As a corollary we show that for any two positive integersk, g, there exists ak-chromatic graph of girth at leastg whose star-chromatic number is alsok. The special case of this corollary withg=4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose star-chromatic number equals their chromatic number. We then study the star-chromatic number of An infinite family of graphs is constructed to show that for each >0 and eachm2 there is anm-connected (m+1)-critical graph with star chromatic number at mostm+. This answers another question asked by Abbott and Zhou.     

On the vertex face total chromatic number of planar graphs

Let G be a planar graph. The vertex face total chromatic number _χ13(G) of G is the least number of colors assigned to V(G)∪F(G) such that no adjacent or incident elements receive the same color. The main results of this paper are as follows: (1) We give the vertex face total chromatic number for all outerplanar graphs and modulus 3-regular maximal planar graphs. (2) We prove that if G is a maximal planar graph or a lower degree planar graph, i.e., ∠(G) ≤ 3, then _χ13(G) ≤ 6. © 1996 John Wiley & Sons, Inc.     

Inequalities for the first‐fit chromatic number

The First‐Fit (or Grundy) chromatic number of G, written as χ_FF(G), is defined as the maximum number of classes in an ordered partition of V(G) into independent sets so that each vertex has a neighbor in each set earlier than its own. The well‐known Nordhaus‐‐Gaddum inequality states that the sum of the ordinary chromatic numbers of an n‐vertex graph and its complement is at most n + 1. Zaker suggested finding the analogous inequality for the First‐Fit chromatic number. We show for n ≥ 10 that ?(5n + 2)/4? is an upper bound, and this is sharp. We extend the problem for multicolorings as well and prove asymptotic results for infinitely many cases. We also show that the smallest order of C₄‐free bipartite graphs with χ_FF(G) = k is asymptotically 2k² (the upper bound answers a problem of Zaker [9]). © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 75–88, 2008     

Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [17] which generalizes the Borsuk–Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K_{⌈t/2⌉,⌊t/2⌋} in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme. * Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846, T046376, AT048826, and NK62321. † Research partially supported by the NSERC grant 611470 and the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846, T046234, AT048826, and NK62321.     

The circular chromatic number of series‐parallel graphs

In this article, we consider the circular chromatic number χ_c(G) of series‐parallel graphs G. It is well known that series‐parallel graphs have chromatic number at most 3. Hence, their circular chromatic numbers are at most 3. If a series‐parallel graph G contains a triangle, then both the chromatic number and the circular chromatic number of G are indeed equal to 3. We shall show that if a series‐parallel graph G has girth at least 2 ⌊(3k − 1)/2⌋, then χ_c(G) ≤ 4k/(2k − 1). The special case k = 2 of this result implies that a triangle free series‐parallel graph G has circular chromatic number at most 8/3. Therefore, the circular chromatic number of a series‐parallel graph (and of a K₄‐minor free graph) is either 3 or at most 8/3. This is in sharp contrast to recent results of Moser [5] and Zhu [14], which imply that the circular chromatic number of K₅‐minor free graphs are precisely all rational numbers in the interval [2, 4]. We shall also construct examples to demonstrate the sharpness of the bound given in this article. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 14–24, 2000     

An approach to hedetniemi's conjecture

For a fixed integer n ? ω, a graph G of chromatic number greater than n is called persistent if for all n + 1-chromatic graphs H, the products G × H are n + 1-chromatic graphs. Wheter all graphs of chromatic number greater than n are persistent is a long-standing open problem due to Hedetniemi. We call a graph G strongly persistent if G is persistent and the Hajos sum of G with any other persistent graph H is still persistent. This paper extends the class of known persistent graphs by proving the following results: If G is constructed from copies of K_n+1 by Hajos sums, adding vertices and edges and at most one contraction of nonadjacent vertices, then G is strongly persistent. © 1929 John Wiley & Sons, Inc.     

平面图的循环色数及临界性

循环着色是普通着色的推广．本文中,我们研究了一类平面图-“花图”的循环着色问题,证明了由2r 1个长为2n 1的圈构成的“辐路”长度为m的花图Fr,m,n的循环色数是2 1/(n-m/2),并证明了在这类图中去掉任何一个点或边后,循环色数都严格减少但普通色数不减少,即这类图是循环色临界的但不是普通色临界的．同时,我们还研究了循环着色与图Gkd中的链之间的关系,给出了两个等价的条件．     

New upper bounds on harmonious colorings

We present an improved upper bound on the harmonious chromatic number of an arbitrary graph. We also consider ?fragmentable”? classes of graphs (an example is the class of planar graphs) that are, roughly speaking, graphs that can be decomposed into bounded-sized components by removing a small proportion of the vertices. We show that for such graphs of bounded degree the harmonious chromatic number is close to the lower bound (2m)^1/2, where m is the number of edges.     

Total chromatic number of complete r-partite graphs

Rosenfeld (1971) proved that the Total Colouring Conjecture holds for balanced complete r-partite graphs. Bermond (1974) determined the exact total chromatic number of every balanced complete r-partite graph. Rosenfeld's result had been generalized recently to complete r-partite graphs by Yap (1989). The main result of this paper is to prove that the total chromatic number of every complete r-partite graph G of odd order is Δ (G) + 1. This result gives a partial generalization of Bermond's theorem.