Neighbor Sum Distinguishing Total Coloring of Triangle Free IC-planar Graphs  

作　　者：Wen Yao SONG Yuan Yuan DUAN Lian Ying MIAO 

机构地区：[1]School of Mathematics and Statistics,Zaozhuang University,Zaozhuang 277160,P.R.C hina [2]School of Mathematics,China University of Mining and Technology,Xuzhou 221116,P.R.C hina

出　　处：《Acta Mathematica Sinica,English Series》2020年第3期292-304,共13页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China(Grant No.11771443);the Foundation of Innovative Science and Technology for Youth in Universities of Shandong Province,China(Grant No.2019KJI001)。

摘　　要：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.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 by χ∑ " is the neighbor sum distinguishing total chromatic number.Pilsniak and Wozniak conjectured χ∑"(G)≤Δ(G)+3 for any simple graph with maximum degree Δ(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 with Δ(G)≥6.

关 键 词：NEIGHBOR SUM distinguishing total COLORING Combinatorial Nullstellensatz TRIANGLE free IC-planar graph MAXIMUM DEGREE 

分 类 号：O157.5[理学—数学]