Neighbor Sum Distinguishing Chromatic Index of Sparse Graphs via the Combinatorial Nullstellensatz

Let Ф ： E（G）→ {1, 2,…, k}be an edge coloring of a graph G. A proper edge-k-coloring of G is called neighbor sum distinguishing if ∑eЭu Ф（e）≠∑eЭu Ф（e） for each edge uv∈E（G）.The smallest value k for which G has such a coloring is denoted by χ＇Σ（G） which makes sense for graphs containing no isolated edge（we call such graphs normal）. It was conjectured by Flandrin et al. that χ＇Σ（G） ≤△（G） ＋ 2 for all normal graphs,except for C5. Let mad（G） = max{（2｜E（H）｜）/（｜V（H）｜）｜HЭG}be the maximum average degree of G. In this paper,we prove that if G is a normal graph with△（G）≥5 and mad（G） 〈 3-2/（△（G））, then χ＇Σ（G）≤△（G） ＋ 1. This improves the previous results and the bound △（G） ＋ 1 is sharp.

Neighbor Distinguishing Total Choice Number of Sparse Graphs via the Combinatorial Nullstellensatz

Let G =（V, E） be a graph and Ф ： V tA E → {1, 2,..., k） be a total-k-coloring of G. Let f（v）（S（v）） denote the sum（set） of the color of vertex v and the colors of the edges incident with v. The total coloring Ф is called neighbor sum distinguishing if （f（u） ≠ f（v）） for each edge uv∈ E（G）. We say that Фis neighbor set distinguishing or adjacent vertex distinguishing if S（u） ≠ S（v） for each edge uv ∈ E（G）. For both problems, we have conjectures that such colorings exist for any graph G if k 〉 △（G） ＋ 3. The maximum average degree of G is the maximum of the average degree of its non-empty subgraphs, which is denoted by mad （G）. In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that these two conjectures hold for sparse graphs in their list versions. More precisely, we prove that every graph G with maximum degree A（G） and maximum average degree mad（G） has ch＇＇∑（G） 〈 △（G） ＋ 3 （where ch＇＇∑（G） is the neighbor sum distinguishing total choice number of G） if there exists a pair （k, m） ∈ {（6, 4）, （5, 18/5）, （4, 16）} such that △（G） 〉 k and mad （G） 〈 m.

A GENERALIZATION OF HILBERT NULLSTELLENSATZ

Ⅰ. RESULTSIn [1], A. Abian proved the following theorem.Theorem A （Abian）. Let C be the complex field, P=C[x0, xi,…, xζ,…] be a polynomial ring over C, where {x0, x1,…, xζ,…} is a set of algebraically independent indeterminates over C with cardinality not exceeding |C|. Let be a subset of P with smaller than |C|. If every finite subset of has a common zero-point in C, then the whole set has a common zero-

一类稀疏图的邻和可区别全染色

利用组合零点定理和权转移法,研究了一类稀疏图的邻和可区别全染色,证明了这类图的邻和可区别全色数不超过Δ+3,得到了邻和可区别全色数猜想对这类稀疏图是成立的。

Neighbor sum distinguishing total colorings via the Combinatorial Nullstellensatz

Let G=(V,E)be a graph andφbe a total coloring of G by using the color set{1,2,...,k}.Let f(v)denote the sum of the color of the vertex v and the colors of all incident edges of v.We say thatφis neighbor sum distinguishing if for each edge uv∈E(G),f(u)=f(v).The smallest number k is called the neighbor sum distinguishing total chromatic number,denoted byχ′′nsd(G).Pil′sniak and Wo′zniak conjectured that for any graph G with at least two vertices,χ′′nsd(G)(G)+3.In this paper,by using the famous Combinatorial Nullstellensatz,we show thatχ′′nsd(G)2(G)+col(G)-1,where col(G)is the coloring number of G.Moreover,we prove this assertion in its list version.

子立方图的2-距离和可区别边色数

2-距离和可区别边色数χ′2−P(G)是指使图G有一个正常边染色且任意一对距离为2的顶点的颜色集合的和不相等的最小颜色数值k.本文证明了:若图G是子立方图,则有χ′2−P(G)≤9.

构造弹性力学位移函数的机械化算法

本文证明线性算子方程组Ａｕ＝ｆ的一般解为ｕ＝Ｃｖ＋ｅ，其中ｖ满足方程组Ｄｖ＝ｇ，Ｄ是对角矩阵。以Ｈｉｌｂｅｒｔ零点定理的构造性证明为基础，给出了Ｃ，Ｄ，ｅ的机械化求示。用此方法可以给出各种弹性力学位移函数的机械化算法。

关于交换环上矩阵的点定理

针对交换环上的矩阵,建立了一些有关特征值的基本结论。作为本文的主要结果,关于交换环上矩阵的正点定理,零点定理和非负点定理被建立。这些定理可看作关于交换环的抽象点定理的一种推广。

关于交换环上矩阵的高层点定理

在关于交换环的高层正点定理和高层非负点定理的基础上,在交换环范畴中进一步建立了所谓的高层零点定理。同时,针对交换环上矩阵,获得了关于交换环上矩阵的高层正点定理,高层零点定理和高层非负点定理。该点定理可看作是关于交换环上矩阵的通常点定理的一种推广。

零点定理在电桥测电阻实验中的应用

通过对电桥测电阻实验中的测量方法的研究,根据测量方法中遇到的检流计电流改变方向及分区间测量的问题,引入高等数学中的零点定理进行叙述,将理论知识与实际操作结合起来,提高对实验及操作的进一步认知。

Neighbor Sum Distinguishing Index of Graphs with Maximum Average Degree

A proper k-edge coloring of a graph G = (V(G), E(G)) is an assignment c: E(G) → {1, 2, …, k} such that no two adjacent edges receive the same color. A neighbor sum distinguishing k-edge coloring of G is a proper k-edge coloring of G such that for each edge uv ∈ E(G). The neighbor sum distinguishing index of a graph G is the least integer k such that G has such a coloring, denoted by χ’Σ(G). Let be the maximum average degree of G. In this paper, we prove χ’Σ(G) ≤ max{9, Δ(G) +1} for any normal graph G with . Our approach is based on the discharging method and Combinatorial Nullstellensatz.

一类函数的几个特殊性质

在函数的零点定理和不动点定理的基础上,定义了函数的准不动点,并证明了一类函数有关零点和准不动点的几个性质.

可数生成代数上的Hilbert零点定理

证明了Hilbert 零点定理可以推广到一个不可数域可数生成的代数上,还给出了它推广后的几何形式.

带有最大平均度限制的图的邻和可区别全可选择数

利用组合零点定理和权转移法,研究了最大度Δ(G)≥8且最大平均度mad(G)<14/3的图G的邻和可区别列表全染色,确定了该类图的邻和可区别全可选择数不超过Δ(G)+3.

围长至少为21的平面图的邻和可区分的顶点列表色数

设f是从图G的顶点集合V到整数集合N的一个映射,令每一个点v的邻和为S_f(v)=Σ_(u∈NG(v))f(u),若f满足任意相邻两点的邻和不相等,则称f是图G的一个邻和可区分的顶点列表标号。设L为图G的一个k-列表配置,对任意点v有f(v)∈L(v)。若存在最小的正整数k使得对任意L,图G都有一个邻和可区分的顶点列表标号f,则称k为图G的邻和可区分的顶点列表色数,η_l(G)。证明当平面图G的围长至少为21时,图G的邻和可区分的顶点列表色数η_l(G)至多为3。

Neighbor Sum Distinguishing Total Choosability of Planar Graphs with Maximum Degree at Least 10

A neighbor sum distinguishing(NSD)total coloringφof G is a proper total coloring of G such thatΣz∈EG(u)U{u}φ(z)≠Σz∈EG(v)U{v}φ(z)for each edge uv∈E(G),where EG(u)is the set of edges incident with a vertex u.In 2015,Pilśniak and Wozniak conjectured that every graph with maximum degreeΔhas an NSD total(Δ+3)-coloring.Recently,Yang et al.proved that the conjecture holds for planar graphs withΔ≥10,and Qu et al.proved that the list version of the conjecture also holds for planar graphs withΔ≥13.In this paper,we improve their results and prove that the list version of the conjecture holds for planar graphs withΔ≥10.

无三角形NIC-可平面图的邻和可区别全可选性

设图G=(V(G),E(G)),Φ为图G的正常k-全染色,令C_(Φ)(v)表示顶点v以及与顶点v关联的边所染颜色的和.若对任意一条边uv∈E(G)都有C_(Φ)(u)≠C_(Φ)(v),则称Φ是图G的邻和可区别k-全染色.图G的全分配L是指对图G的每个元素z∈V(G)∪E(G)分配一个颜色集合L(z).如果对图G的任意全分配L={L(z)||L(z)|≥k,z∈(G)∪E(G)},图G都存在邻和可区别k-全染色Φ,使得任意z∈V(G)∪E(G),都有Φ(z)∈L(z),则称图G是邻和可区别k-全可选的.使得图G是邻和可区别k-全可选的最小正整数k称为图G的邻和可区别全可选性,记为ch_(∑)″(G).本文运用组合零点定理和权转移方法研究了无三角形NIC-可平面图的邻和可区别全可选性,证明了若图G为无三角形NIC-可平面图,则ch_(∑)″(G)≤max{Δ(G)+3,15}.

无限维空间中的实零点定理

在本文中，我们建立了无限维空间中的实零点定理，同时从仿射空间的拓扑结构和域的序结构两个方面，分别刻划了适合无限维实零点定理的序域．此外，本文有例子表明，对任意的基数α，确实存在适合α维实零点定理的序域．

Neighbor Sum Distinguishing Total Colorings of Triangle Free Planar Graphs

A total k-coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2,...,k}. Let f（u） denote the sum of the color on a vertex u and colors on all the edges incident to u. A k-neighbor sum distinguishing total coloring of G is a total k-coloring of G such that for each edge uv ∈ E（G）, f（u） ≠ f（v）. By X＂nsd（G）, we denote the smallest value k in such a coloring of G. Pilgniak and Wozniak conjectured that X＂nsd（G） ≤ △（G）＋ 3 for any simple graph with maximum degree △（G）. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that the conjecture holds for any triangle free planar graph with maximum degree at least 7.

Neighbor Sum Distinguishing Colorings of Graphs with Maximum Average Degree Less Than 37/12

Let G be a graph and let its maxiraum degree and maximum average degree be denoted by △（G） and mad（G）, respectively. A neighbor sum distinguishing k-edge colorings of graph G is a proper k-edge coloring of graph G such that, for any edge uv ∈ E（G）, the sum of colors assigned on incident edges of u is different from the sum of colors assigned on incident edges of v. The smallest value of k in such a coloring of G is denoted by X∑ （G）. Flandrin et al. proposed the following conjecture that X＇∑ （G） ≤△ （G） ＋ 2 for any connected graph with at least 3 vertices and G ≠ C5. In this paper, we prove that the conjecture holds for a normal graph with mad（G） 〈 37/12and △ （G）≥ 7.