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 Neighbor Distinguishing Total Choice Number of Sparse Graphs via the Combinatorial Nullstellensatz Qu, Cun-quan1; Ding, Lai-hao1; Wang, Guang-hui1; Yan, Gui-ying2 2016-06-01 Source Publication ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES ISSN 0168-9673 Volume 32Issue:2Pages:537-548 Abstract Let G = (V, E) be a graph and phi : V boolean OR 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 phi is called neighbor sum distinguishing if (f(u) not equal f(v)) for each edge uv is an element of E(G). We say that phi is neighbor set distinguishing or adjacent vertex distinguishing if S(u) not equal S(v) for each edge uv is an element of E(G). For both problems, we have conjectures that such colorings exist for any graph G if k >= Delta (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 Delta(G) and maximum average degree mad(G) has ch(Sigma)''(G) <= Delta(G) + 3 (where ch(Sigma)'' (G) is the neighbor sum distinguishing total choice number of G) if there exists a pair (k, m) is an element of{(6, 4), (5, 18/5), (4, 16/5)} such that Delta(G) >= k and mad (G) < m. Keyword neighbor sum distinguishing total coloring Combinatorial Nullstellensatz neighbor sum distinguishing total choice number DOI 10.1007/s10255-016-0583-8 Language 英语 Funding Project National Natural Science Foundation of China ; National Natural Science Foundation of China ; Foundation for Distinguished Young Scholars of Shandong Province[JQ201501] ; Natural Science Foundation of Shandong Province[ZR2013AM001] ; Independent Innovation Foundation of Shandong University[IFYT14012] ; Fundamental Research Funds of Shandong University WOS Research Area Mathematics WOS Subject Mathematics, Applied WOS ID WOS:000375233700028 Publisher SPRINGER HEIDELBERG Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/22581 Collection 应用数学研究所 Corresponding Author Wang, Guang-hui Affiliation 1.Shandong Univ, Sch Math, Jinan 250100, Peoples R China2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China Recommended CitationGB/T 7714 Qu, Cun-quan,Ding, Lai-hao,Wang, Guang-hui,et al. Neighbor Distinguishing Total Choice Number of Sparse Graphs via the Combinatorial Nullstellensatz[J]. ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES,2016,32(2):537-548. APA Qu, Cun-quan,Ding, Lai-hao,Wang, Guang-hui,&Yan, Gui-ying.(2016).Neighbor Distinguishing Total Choice Number of Sparse Graphs via the Combinatorial Nullstellensatz.ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES,32(2),537-548. MLA Qu, Cun-quan,et al."Neighbor Distinguishing Total Choice Number of Sparse Graphs via the Combinatorial Nullstellensatz".ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES 32.2(2016):537-548.
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