Injective (Δ + 1)-coloring of planar graphs with girth 6

A vertex coloring of a graph G is called injective if every two vertices joined by a path of length 2 get different colors. The minimum number χ _i (G) of the colors required for an injective coloring of a graph G is clearly not less than the maximum degree Δ(G) of G. There exist planar graphs with girth g ≥ 6 and χ _i = Δ+1 for any Δ ≥ 2. We prove that every planar graph with Δ ≥ 18 and g ≥ 6 has χ _i ≤ Δ + 1.     

List 2-distance (Δ + 1)-coloring of planar graphs with girth at least 7

The trivial lower bound for the 2-distance chromatic number χ ₂(G) of a graph G with maximum degree Δ is Δ + 1. There are available some examples of the graphs with girth g ≤ 6 that have arbitrarily large Δ and χ ₂(G) ≥ Δ + 2. In the paper we improve the known restrictions on Δ and g under which a planar graph G has χ ₂(G) = Δ + 1.     

List 2-distance (Δ + 2)-coloring of planar graphs with girth 6 and Δ ≥ 24

It was proved in [1] that every planar graph with girth g ≥ 6 and maximum degree Δ ≥ 8821 is 2-distance (Δ + 2)-colorable. We prove that every planar graph with g ≥ 6 and Δ ≥ 24 is list 2-distance (Δ + 2)-colorable.     

Corrigendum to “(p,1)-total labelling of graphs” [Discrete Math. 308 (2008) 496–513]

We give a corrected proof of Proposition 51 of the paper cited in the title.     

P_(4k-1)-factorization of complete bipartite graphs

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for the existence of Pv-factorization of Km,n.When v is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for v = 3. In this paper we will show that Ushio Conjecture is true when v = 4k - 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P4k-1-factorization of Km,n is (1) (2k - 1)m ≤ 2kn, (2) (2k -1)n≤2km, (3) m n ≡ 0 (mod 4k - 1), (4) (4k -1)mn/[2(2k -1)(m n)] is an integer.     

Evaluation of Series Involving the Product of the Tail of $${\zeta(k)}$$ and $${\zeta(k+1)}$$ζ(k+1)

We aim at evaluating the following class of series involving the product of the tail of two consecutive zeta function values

$$\sum\limits_{n=1}^{\infty}\left(\zeta(k)-1-\frac{1}{2^k}-\cdots-\frac{1}{n^k}\right)\left(\zeta(k+1)-1-\frac{1}{2^{k+1}}-\cdots-\frac{1}{n^{k+1}}\right),$$

where \({k\geq 2}\) is an integer. We show that the series can be expressed in terms of Riemann zeta function values and a special integral involving a polylogarithm function.

     

关于sum from k=1 to ∞ f(k)x~(k-1)计算的矩阵方法

本文将幂级数 ∑∞k=1f (k) xk- 1的计算转化为求解矩阵方程 Aβ =α,使计算非常简单和有效 .     

Plane graphs of maximum degree Δ ≥ 7 are edge-face (Δ + 1)-colorable

A plane graph G is edge-face k-colorable if its edges and faces can be colored with k colors such that any two adjacent or incident elements receive different colors. It is known that every plane graph G of maximum degree Δ ≥ 8 is edge-face (Δ + 1)-colorable. The condition Δ ≥ 8 is improved to Δ ≥ 7 in this paper.     

Skew-symmetric association schemes with two classes and strongly regular graphs of type L 2n−1(4n−1)

A construction of a pair of strongly regular graphs ⁿ and ⁿ of type L _2n–1(4n–1) from a pair of skew-symmetric association schemes W, W of order 4n–1 is presented. Examples of graphs with the same parameters as ⁿ and ⁿ, i.e., of type L _2n–1(4n–1), were known only if 4n–1=p ³, where p is a prime. The first new graph appearing in the series has parameters (v, k, )=(225, 98, 45). A 4-vertex condition for relations of a skew-symmetric association scheme (very similar to one for the strongly regular graphs) is introduced and is proved to hold in any case. This has allowed us to check the 4-vertex condition for ⁿ and ⁿ, thus to prove that ⁿ and ⁿ are not rank three graphs if n>2.     

用待定系数法求sum from k=1 to n(f(k)q~(k-1))

设f(x)=a_0+a_1x+a_2x~2_…+a_mx~m,其中a_0,a_1,…,a_m为常数,a_m(?)0,m≥0。定理1 若q=1,则存在常数项为零的m+1次多项式g(x),使得     

The series sum from(n=1) to ∞(1/(n~(k 1))e(-z~(2k))/(n~(2k)))(oddk) and Riemann Zeta function

J.Tennenbaum discussed the function sum from n=1 to ∞() 1/n~2 e~(-2/n) in 1977.Zhang Nanyue discussed the function sum from n=1 to 1 () 1/n~2e~(-z~2/n~2) in 1983.Now we discuss the functions sum from n=1 to ∞ () 1/n~(k 1).e~(z~(2k)/n~(2k))(kpositive odd)in this paper which finds representations of two integrales about Riemann Zeta function     

关于(4k+1)连通的(8k+6)维闭光滑流形的同胚分类

本文给出了光滑流形S^4k+2×S^4k+3(k≥1)上自微分同胚拟同痕的充分必要条件,并且计算了这些拟同痕类全体按复合作乘法所构成的群Π₀Diff#_m¹S^4k+2×S^4k+3,作为这些结果的一个直接应用,我们对一类(4k+1)连通的(8k+6)维闭光滑流形做了完全分类。     

A Totally(Δ+1)-colorable 1-planar Graph with Girth at Least Five

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge.In this paper,we prove that every 1-planar graph G with maximum degree Δ(G)≥12 and girth at least five is totally(Δ(G)+1)-colorable.     

用求和矩阵求sum from k=1 to n(K~m)

若将上面六个关于n的多项式的系数,依次作为矩阵的第一列、第二列、…、第六列,缺元素处补以0,则构成一个矩阵     

A remark on the intersection arrays of distance regular graphs and the distance regular graphs of diameter d = 3i − 1 with b i = 1 and k > 2

Let be a distance regular graph with intersection array b ₀, b ₁,..., b _d–1; c ₁,..., c _d. It is shown that in some cases (c _i–1, a _i–1, b _i–1) = (c ₁, a ₁, b ₁)and (c _2i–1, a _2i–1, b _2i–1) imply k 2b _i + 1. As a corollary all distance regular graphs of diameter d = 3i – 1 with b _i = 1 and k > 2 are determined.     

欧拉公式sum from k=1 to n 1/k=lnn+c+r_n(r_n→0)中余项r_n的估计

<正> 本文首先推导出不等式、由不等式(1)推导出r_n的估计式为:从这一结果可推出     

On the Diophantine Equation sum from k=1 to m((k~n)=(m+1)~n)

P. Erds has conjectured [1] that the Diophantine equation 1~n+2~n+…+m~n=(m+1)~n (1) has no positive integer solutions except that n=1, m=2. It is true when m≤10~(10) [3]. A generalized form of (1) has been investigated in [1] [2], and various