A proper edge-k-coloring of a graph G is a mapping from E（G） to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv ∈ E（G）, the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. Let X（G ） denote the smallest value k in such a ＇ G coloring of G. This parameter makes sense for graphs containing no isolated edges （we call such graphs normal）. The maximum average degree mad（G） of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad（G） 〈 5 then x＇（G） ≤ 5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.
Xiao Wei YUGuang Hui WANGJian Liang WUGui Ying YAN
An acyclic edge coloring of a graph is a proper edge coloring such that every cycle contains edges of at least three distinct colors. The acyclic chromatic index of a graph G, denoted by a＇（G）, is the minimum number k such that there is an acyclic edge coloring using k colors. It is known that a＇（G） ≤ 16△ for every graph G where △denotes the maximum degree of G. We prove that a＇（G） 〈 13.8A for an arbitrary graph G. We also reduce the upper bounds of a＇（G） to 9.8△ and 9△ with girth 5 and 7, respectively.
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.
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.
An r-acyclic edge chromatic number of a graph G,denoted by α＇r（G）,is the minimum number of colors used to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min {｜C｜,r} colors.We prove that α＇r（G） ≤（4r ＋ 1）△（G）,when the girth of the graph G equals to max{50,△A（G）} and 4 ≤ r ≤ 7.If we relax the restriction of the girth to max {220,A（G）},the upper bound of a＇r（G） is not larger than（2r ＋ 5）△（G） with 4 ≤r≤ 10.
