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| 1 | Neighbor Sum Distinguishing Total Colorings of Triangle Free Planar Graphs显示文摘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 χ nsd(G), we denote the smallest value k in such a coloring of G. Pil′sniak and Wo′zniak conjectured that χ nsd(G) ≤Δ(G) + 3 for any simple graph with maximum degree Δ(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. | Ji Hui WANG Qiao Ling MA Xue HAN | 2015 | Acta Mathematica Sinica,English Series2015，31，2： | 4 |
| 2 | Neighbor Distinguishing Total Choice Number of Sparse Graphs via the Combinatorial Nullstellensatz显示文摘Let G =(V,E) be a graph and φ:V∪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 φis called neighbor sum distinguishing if(f(u)≠f(v)) for each edge uv ∈ E(G).We say that φ is 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 △(G) and maximum average degree mad(G) has ch_Σ '(G) ≤△(G) + 3(where ch_Σ'(G) is the neighbor sum distinguishing total choice number of G) if there exists a pair(k,m) ∈ {(6,4),(5,18/5),(4,16/5)}such that △(G) ≥ k and mad(G) < m.neighbor sum distinguishing total choice number of G) if there exists a pair(k,m) ∈ {(6,4),(5,18/5),(4,16/5)}such that △(G)≥k and mad(G) < m. | Cun-quan QU Lai-hao DING Guang-hui WANG Gui-ying YAN | 2016 | Acta Mathematicae Applicatae Sinica2016，32，2： | 2 |
| 3 | △=3的图的邻和可区别全可选性(英文)显示文摘设图G=(V,E),φ:V∪E→{1,2,…,k}为图G的一个正常全染色.令f(v)表示点v及所有与其关联的边的颜色的加和.若对任意uv∈E(G),有f(u)≠f(v),则称φ是图G的邻和可区别全染色.Pilsniak和Wozniak最早研究了邻和可区别全染色,并猜想对于任意图G,若k≥△(G)+3,则其存在邻和可区别全染色.图G的最大平均度,记为mad(G),是G的所有非空子图的平均度的最大值.本文运用组合零点定理与权转移方法证明了:若图G满足△(G)=3且mad(G)<(44)/(15),则ch_Σ″(G)≤6(其中ch_Σ″(G)为图G的邻和可区别全可选性). | 姚京京 邵泽玲 徐常青 | 2016 | 数学进展2016，45，3： | 2 |
| 4 | Neighbor sum distinguishing total chromatic number of K4-minor free graph显示文摘图 G 的 k 全部的着色是印射:V (G) E (G){ 1;2, ... , k } 以便没有二邻近或在 V (G) E (G) 的事件元素收到一样的颜色。让 f (v) 与 v 在所有边事件上在顶点 v 和颜色上表示颜色的和:我们说那是区分 G 的全部的着色的 k 邻居和如果 f (u) 为每边 uv E (G) 的 6 f (v) :表示 (G) 在 G 的如此的着色的最小的价值 k:Pilniak 和 Woniak 与最大的度(G) 为任何简单的图推测了那, (G)+3。在这份报纸,由使用著名组合 Nullstellensatz,我们为 K 4-minor 证明那有(G) 的免费的图 G > 5;=(G)+ 1 如果 G 不不那样包含二邻近的 -vertices, , (G)=(G)+ 2。 | Hongjie SONG Changqing XU | 2017 | Frontiers of Mathematics in China2017，12，4： | 1 |
| 5 | 双圈图的邻和可区别边染色显示文摘设G是阶数不小于3的简单连通图.u,v是图G的一个k-正常边染色的任意相邻的两个顶点,如果点u所有关联边的颜色加和与点v所有关联边的颜色加和不相等,则称该染色是邻和可区别的.对G进行邻和可区别边染色所需要的最少的颜色数k称为G的邻和可区别边色数.根据双圈图的结构特点,对双圈图的有根树的树高进行分类,运用结构分析法、反证法、构造染色法,以及组合零点定理等方法,研究了双圈图的邻和可区别边染色问题,得到了双圈图的邻和可区别边色数. | 谭钧铭 强会英 刘欢 王洪申 | 2022 | 西南大学学报(自然科学版)2022，44，6： | 0 |
| 6 | Neighbor Sum Distinguishing Total Coloring of Triangle Free IC-planar Graphs显示文摘A graph is IC-planar if it admits a drawing in the plane such that each edge is crossed at most once and two crossed edges share no common end-vertex.A proper total-k-coloring of G is called neighbor sum distinguishing if∑_c(u)≠∑_c(v)for each edge uv∈E(G),where∑_c(v)denote the sum of the color of a vertex v and the colors of edges incident with v.The least number k needed for such a total coloring of G,denoted byχ∑'is the neighbor sum distinguishing total chromatic number.Pilsniak and Wozniak conjecturedχ∑'(G)≤Δ(G)+3 for any simple graph with maximum degreeΔ(G).By using the famous Combinatorial Nullstellensatz,we prove that above conjecture holds for any triangle free IC-planar graph with△(G)≥7.Moreover,it holds for any triangle free planar graph withΔ(G)≥6. | Wen Yao SONG Yuan Yuan DUAN Lian Ying MIAO | 2020 | Acta Mathematica Sinica,English Series2020，36，3： | 0 |
| 7 | Neighbor Sum Distinguishing Total Choosability of Planar Graphs with Maximum Degree at Least 10显示文摘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. | Dong-han Zhang You Lu Sheng-gui Zhang Li Zhang | 2024 | Acta Mathematicae Applicatae Sinica2024，40，1： | 0 |
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