完全3-部图K_(1，10，n)的交叉数

在上世纪五十年代初,Zarankiewicz猜想完全2-部图Km,m(m≤n)的交叉数为[m/2][m-1/2][n/2][n-1/2](对任意实数x,[x]表示不超过x的最大整数),目前只证明了当m ≤ 6时,Zarankiewicz猜想是正确的.假定Zarankiewicz猜想对m=11的情形成立,本文确定完全3-部图K1,10,n的交叉数.     

The graph Ramsey number R(Fℓ,K6)

For a given pair of two graphs $(F,H)$, let $R(F,H)$ be the smallest positive integer $r$ such that for any graph $G$ of order $r$, either $G$ contains $F$ as a subgraph or the complement of $G$ contains $H$ as a subgraph. Baskoro, Broersma and Surahmat (2005) conjectured that $R(F_{\ell },K_n)=2\ell (n-1)+1$for $\ell \ge n\ge 3$, where $F_{\ell }$ is the join $K_1+\ell K_2$ of $K_1$ and $\ell K_2$. In this paper, we prove that this conjecture is true for the case $n=6$.     

Multicolor bipartite Ramsey number of C4 and large Kn,n

Let br_k(C₄;K_{n, n}) be the smallest N such that if all edges of K_{N, N} are colored by k + 1 colors, then there is a monochromatic C₄ in one of the first k colors or a monochromatic K_{n, n} in the last color. It is shown that br_k(C₄;K_{n, n}) = Θ(n²/log²n) for k?3, and br₂(C₄;K_{n, n})≥c(n n/log²n)² for large n. The main part of the proof is an algorithm to bound the number of large K_{n, n} in quasi‐random graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 47‐54, 2011     

All cycle‐complete graph Ramsey numbers r(Cm,K6)

The cycle‐complete graph Ramsey number r(C_m, K_n) is the smallest integer N such that every graph G of order N contains a cycle C_m on m vertices or has independence number α(G) ≥ n. It has been conjectured by Erd?s, Faudree, Rousseau and Schelp that r(C_m, K_n) = (m ? 1) (n ? 1) + 1 for all m ≥ n ≥ 3 (except r(C₃, K₃) = 6). This conjecture holds for 3 ≤ n ≤ 5. In this paper we will present a proof for n = 6 and for all n ≥ 7 with m ≥ n² ? 2n. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 251–260, 2003     

素数阶循环图和经典Ramsey数R（4，n）的三个新下界

研究了素数阶循环圈的基本性质，提出了寻求有效参数构造正则循环圈的新方法，得到了3个经典Ramsey数的新下界：R（4，17）≥164，R（4，18）≥182，R（4，22）≥282．这前2个结果填补了关于Ramsey数综述[2]的上下界表中的2个空白，第3个结果超过了目前已知的最好下界R（4，22）≥258，     

关于二部图K(m,n)-2的色唯一性

设Ｋ（ｍ，ｎ）－２表示从完全二部图Ｋ（ｍ，ｎ）中删去任意２条边所得之图．本文证明了：１．若ｎ≥ｍ≥３，且ｎ＋ｍ＞(（ｎ－ｍ）^２＋８)^1/2＋１/２（ｎ－ｍ）^２＋４，则Ｋ（ｍ，ｎ）－２是色唯一图；２．当ｍ≥３时，Ｋ（ｍ，ｍ）－２，Ｋ（ｍ，ｍ＋１）－２和Ｋ（ｍ，ｍ＋２）－２均是色唯一图．     

完全二部图K_(1,n),K_(2,n)和K_(3,n)的点强可区别全染色

设f是图G的一个正常全染色.对任意x∈V(G),令C(x)表示与点x相关联或相邻的元素的颜色以及点x的颜色所构成的集合.若对任意u,v∈V(G),u≠v,有C(u)≠C(v),则称.f是图G的一个点强可区别全染色,对一个图G进行点强可区别全染色所需的最少的颜色的数目称为G的点强可区别全色数,记为X_(vst)(G).讨论了完全二部图K_(1,n),K_(2,n)和L_(3,n)的点强可区别全色数,利用组合分析法,得到了当n≥3时,X_(vst)(K_(1,n)=n+1,当n≥4时,X_(vst)(K_(2,n)=n+2,当n≥5时,X_(vst)(K_(3,n))=n+2.     

关于K2,s+1 VS,K1,n的Ramsey数

It is shown that the Ramsey number r(K2,s 1, K1,n) ≤ n √sn (s 3)/2 o(1) for large n, and r(K2,s 1, K1,n)∈{(q-1)2/s 1,-(q-1)2/s 2},wheren (q-1)2/s-q 2 and q is a prime power such that s|(q - 1).     

New Lower Bounds on the Multicolor Ramsey Numbers R r (C 2 m )

The multicolor Ramsey number R_r(H) is defined to be the smallest integer n=n(r) with the property that any r-coloring of the edges of the complete graph K_n must result in a monochromatic subgraph of K_n isomorphic to H. It is well known that 2^rm<R_r(C_2m+1)<2(r+2)!m and R_r(C_2m)≥(r−1)(m−1)+1. In this paper, we prove that R_r(C_2m)≥2(r−1)(m−1)+2. This research is supported by NSFC(60373096, 60573022) and SRFDP(20030141003)     

关于完全三部图K(m,n,r)的色唯一性

本文研究完全三部图K(m,n,r)的色唯一性问题，通过比较两个色等价图的色划分数的方法，得出两个关于K(m,n,r)为色唯一图的一般形式数值条件，基本上解决了K(m,n,r)为色唯一图的判定问题．     

关于完全三部图K(n,n,n+4)的色唯一性

通过比较两个图的色多项式的系数(本文使用了五独立集数)、顶点集、边集、三角形和四圈的个数,证明了K(2,2,6)是色唯一图,从而部分地回答了文[5],[7]中遗留的一个问题,并得到图K(n,n,n 4)(n=2或n 4)是色唯一的.     

广义图K(n,m)的全色数

1965年,M.Behzad和Vizing分别提出了著名的全着色猜想:即对于简单图G有:XT(G)≤△+2,其中△是图G的最大度.本文确定了完全图Kn的广义图K(n,m)的全色数,并利用它证明了Lm×Kn(m≥3)是第Ⅰ型的.     

Remarks on Vertex-Distinguishing IE-Total Coloring of Complete Bipartite Graphs K4,n and Kn,n

Let G be a simple graph.An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color.Let C(u) be the set of colors of vertex u and edges incident to u under f.For an IE-total coloring f of G using k colors,if C(u)=C(v) for any two different vertices u and v of V(G),then f is called a k-vertex-distinguishing IE-total-coloring of G,or a k-VDIET coloring of G for short.The minimum number of colors required for a VDIET coloring of G is denoted by χ ie vt (G),and it is called the VDIET chromatic number of G.We will give VDIET chromatic numbers for complete bipartite graph K4,n (n≥4),K n,n (5≤ n ≤ 21) in this article.     

7个经典Ramsey数R(k,l)的新下界

利用构造性的方法,得到7个经典Ramsey数的新下界R(3,29)≥174,R(4,23)≥272,R(5,24)≥488,R(7,12)≥312,R(8,18)≥728,R(8,20)≥860,R(9,21)≥1278.     

图K(r,2)的邻强边色数

本文给出了每部有2个点的完全r-部图(r≥2)的邻强边色数.     

The smallest surface that contains all signed graphs on K4,n

Finding the smallest number of crosscaps that suffice to orientation-embed every edge signature of the complete bipartite graph $K_{m,n}$ is an open problem. In this paper that number for the complete bipartite graph $K_{4,n}$, $n\ge 4$, is determined by using diamond products of signed graphs. The number is $2?\frac{n?1}{2}?+1$, which is attained by $K_{4,n}$ with exactly 1 negative edge, except that when $n=4$, the number is 4, which is attained by $K_{4,4}$ with exactly 4 independent negative edges.     

关于完全3-部图K1,6,n的交叉数

早在上世纪五十年代,Zarankiewicz猜想完全2-部图Km,n(m≤n)的交叉数为[m/2][m-1/2][n/2][n-1/2](对任意实数x,[x]表示不超过x的最大整数)．目前这一猜想的正确只证明了当m≤6时成立．本文主要证明了若Zarankiewicz猜想对m=7成立,则完全3-部图K1,6,n的交叉数为9[n/2][n-1/2] 6[n/2]．     

On three color Ramsey numbers R(C4,C4,K1,n)

For $k$ given graphs $G_1,G_2,\dots ,G_k$, $k\ge 2$, the $k$-color Ramsey number, denoted by $R(G_1,G_2,\dots ,G_k)$, is the smallest integer $N$ such that if we arbitrarily color the edges of a complete graph of order $N$ with $k$ colors, then it always contains a monochromatic copy of $G_i$ colored with $i$, for some $1\le i\le k$. Let $C_m$ be a cycle of length $m$ and $K_{1,n}$ a star of order $n+1$. In this paper, firstly we give a general upper bound of $R(C_4,C_4,\dots ,C_4,K_{1,n})$. In particular, for the 3-color case, we have $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+3$ and this bound is tight in some sense. Furthermore, we prove that $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+2$ for all $n={?}^2??$ and $?\ge 2$, and if $?$ is a prime power, then the equality holds.     

完全三部图K(n-k,n,n)的色唯一性

通过对图的特征子图个数的比较,给出了图K(n-k,n,n)色唯一性的数值条件.