Neighbor Sum Distinguishing Total Colorings of Graphs with Bounded Maximum Average Degree

A proper [h]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [h] ={1, 2,..., h}. Let w（u） denote the sum of the color on a vertex u and colors on all the edges incident to u. For each edge uv ∈ E（G）, if w（u） ≠ w（v）, then we say the coloring c distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing [h]-total coloring of G. By tndi∑ （G）, we denote the smallest value h in such a coloring of G. In this paper, we obtain that G is a graph with at least two vertices, if mad（G） 〈 3, then tndi∑ （G） ≤k ＋ 2 where k = max{△（G）, 5}. It partially confirms the conjecture proposed by Pilgniak and Wolniak.

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.

Equitable Total Coloring of F_n ∨ W_n

The minimum number of total independent partition sets of V ∪ E of graph G（V,E） is called the total chromatic number of G denoted by χt（G）. If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one＇, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χet（G）. In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order.

A Note on the Minimum Total Coloring of Planar Graphs

Graph coloring is an important tool in the study of optimization, computer science, network design, e.g., file transferring in a computer network, pattern matching, computation of Hessians matrix and so on. In this paper, we consider one important coloring, vertex coloring of a total graph, which is also called total coloring. We consider a planar graph G with maximum degree △（G） 〉 8, and proved that if G contains no adjacent i,j-cycles with two chords for some i,j E {5,6,7}, then G is total-（△ ＋ 1）-colorable.

Acyclic Total Colorings of Planar Graphs without l Cycles

A proper total coloring of a graph G such that there are at least 4 colors on those vertices and edges incident with a cycle of G, is called acyclic total coloring. The acyclic total chromatic number of G is the least number of colors in an acyclic total coloring of G. In this paper, it is proved that the acyclic total chromatic number of a planar graph G of maximum degree at least k and without 1 cycles is at most △（G） ＋ 2 if （k, l） ∈ {（6, 3）, （7, 4）, （6, 5）, （7, 6）}.

List Total Colorings of Planar Graphs without Triangles at Small Distance

Suppose that G is a planar graph with maximum degree △. In this paper it is proved that G is total-（△ ＋ 2）-choosable if （1） △ ≥ 7 and G has no adjacent triangles （i.e., no two triangles are incident with a common edge）; or （2） △ ≥6 and G has no intersecting triangles （i.e., no two triangles are incident with a common vertex）; or （3） △ ≥ 5, G has no adjacent triangles and G has no k-cycles for some integer k ∈ {5, 6}.

On Total Colorings of Some Special 1-planar Graphs

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we verify the total coloring conjecture for every 1-planar graph G if either △（G） ≥9 and g（G）≥ 4, or △（G） ≥ 7 and g（G）≥5, where △（G） is the maximum degree of G and g（G） is the girth of G.

Total Coloring of G×P_n and G×C_n

It is proved that if G is a (+1)-colorable graph, so are the graphs G×Pn and C×Cn, where Pn and Cn are respectively the path and cycle with n vertices, and the maximum edge degree of the graph. The exact chromatic numbers of the product graphs and are also presented. Thus the total coloring conjecture is proved to be true for many other graphs.

Total Coloring of Planar Graphs without Chordal 7-cycles

A k-total-coloring of a graph G is a coloring of vertices and edges of G using k colors such that no two adjacent or incident elements receive the same color.In this paper,it is proved that if G is a planar graph with Δ（G） ≥ 7 and without chordal 7-cycles,then G has a（Δ（G） ＋ 1）-total-coloring.

A Note on List Edge and List Total Coloring of Planar Graphs without Adjacent Short Cycles

LetGbe a planar graph with maximum degreeΔ.In this paper,we prove that if any4-cycle is not adjacent to ani-cycle for anyi∈{3,4}in G,then the list edge chromatic numberχl（G）=Δand the list total chromatic numberχl（G）=Δ＋1.

An Upper Bound for the Adjacent Vertex-Distinguishing Total Chromatic Number of a Graph

Let G = （V, E） be a simple connected graph, and ｜V（G）｜ ≥ 2. Let f be a mapping from V（G） ∪ E（G） to {1,2…, k}. If arbitary uv ∈ E（G）,f（u） ≠ f（v）,f（u） ≠ f（uv）,f（v） ≠ f（uv）; arbitary uv, uw ∈ E（G）（v ≠ w）, f（uv） ≠ f（uw）;arbitary uv ∈ E（G） and u ≠ v, C（u） ≠ C（v）, whereC（u）={f（u）}∪{f（uv）｜uv∈E（G）}.Then f is called a k-adjacent-vertex-distinguishing-proper-total coloring of the graph G（k-AVDTC of G for short）. The number min{k｜k-AVDTC of G} is called the adjacent vertex-distinguishing total chromatic number and denoted by χat（G）. In this paper we prove that if △（G） is at least a particular constant and δ ≥32√△ln△, then χat（G） ≤ △（G） ＋ 10^26 ＋ 2√△ln△.

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.

Neighbor sum distinguishing total chromatic number of K4-minor free graph

A k-total coloring of a graph G is a mapping φ： V（G） U E（G） → {1, 2,..., k} such that no two adjacent or incident elements in V（G） U E（G） receive the same color. Let f（v） denote the sum of the color on the vertex v and the colors on all edges incident with v. We say that ~ is a k-neighbor sum distinguishing total coloring of G if f（u） ≠ f（v） for each edge uv C E（G）. Denote X＂ （G） the smallest value k in such a coloring of G. Pilgniak and Wo/niak conjectured that for any simple graph with maximum degree △（G）, X＂（G） ≤ 3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for Ka-minor free graph G with △（G） ≥ 5, X＂（G） = △（G） ＋ 1 if G contains no two adjacent A-vertices, otherwise, X＂（G） = △（G） ＋ 2.

A Totally(Δ+1)-colorable 1-planar Graph with Girth at Least Five

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we prove that every 1-planar graph G with maximum degree △（G） 〉 12 and girth at least five is totally （△（G）＋1）-colorable.

A Sufficient Condition for Planar Graphs with Maximum Degree 8 to Be 9-totally Colorable

A total k-coloring of a graph G is a coloring of V（G） ∪ E（G） using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number χ＇＇（G） is the smallest integer k such that G has a total k-coloring. It is known that if a planar graph G has maximum degree △≥ 9, then ）χ″（G） =△＋ 1. In this paper, we prove that if O is a planar graph with maximum degree 8 and without a fan of four adjacent 3-cycles, then χ″（G） =- 9.

AVDTC Numbers of Generalized Halin Graphs with Maximum Degree at Least 6

In a paper by Zhang and Chen et al.（see [11]）, a conjecture was made concerning the minimum number of colors Xat（G） required in a proper total-coloring of G so that any two adjacent vertices have different color sets, where the color set of a vertex v is the set composed of the color of v and the colors incident to v. We find the exact values of Xat（G） and thus verify the conjecture when G is a Generalized Halin graph with maximum degree at least 6, A generalized Halin graph is a 2-connected plane graph G such that removing all the edges of the boundary of the exterior face of G （the degrees of the vertices in the boundary of exterior face of G are all three） gives a tree.

图P_m×P_n,P_m×C_n和C_m×C_n的邻点可区别全色数的注记(英文)

Let G be a simple graph. Let f be a mapping from V (G) ∪ E(G) to {1,2,...,k}. Let Cf(v) = {f(v)} ∪ {f(vw)|w ∈ V (G),vw ∈ E(G)} for every v ∈ V (G). If f is a k-proper- total-coloring, and for u,v ∈ V (G),uv ∈ E(G), we have Cf(u) = Cf(v), then f is called a k- adjacent-vertex-distinguishing total coloring (k-AV DTC for short). Let χat(G) = min{k|G have a k-adjacent-vertex-distinguishing total coloring}. Then χat(G) is called the adjacent-vertex- distinguishing total chromatic number (AV DTC number for short). The AV DTC numbers for Pm × Pn,Pm × Cn and Cm × Cn are obtained in this paper.

On Defected Colourings of Graphs

Abstract A k-edge-coloring f of a connected graph G is a （A1, A2, , A）-defected k-edge-coloring if there is a smallest integer/ with 1 _ /3 _〈 k - i such that the multiplicity of each color j E {1,2,... ,/3} appearing at a vertex is equal to Aj _〉 2, and each color of {/3 -}- 1,/3 - 2, - , k} appears at some vertices at most one time. The （A1, A2,, A/）-defected chromatic index of G, denoted as X （A1, A2,, A/; G）, is the smallest number such that every （A1,A2,-.., A/）-defected t-edge-coloring of G holds t _〉 X（A1, A2 A;; G）. We obtain A（G） X（A1, ）2, , A/; G） ＋ -- （Ai - 1） _〈 /k（G） 1, and introduce two new chromatic indices of G i=1 as： the vertex pan-biuniform chromatic index X pb （G）, and the neighbour vertex pan-biuniform chromatic index Xnpb（G）, and furthermore find the structure of a tree T having X pb （T） =1.

若干联图的均匀全染色(英文)

The total chromatic number χt(G) of a graph G(V,E) is the minimum number of total independent partition sets of V E, satisfying that any two sets have no common element. If the difference of the numbers of any two total independent partition sets of V E is no more than one, then the minimum number of total independent partition sets of V E is called the equitable total chromatic number of G, denoted by χet(G). In this paper, we have obtained the equitable total chromatic number of Wm Kn, Fm Kn and Sm Kn while m ≥ n ≥ 3.

Rapid identification of two-dimensional materials via machine learning assisted optic microscopy

A combination of Fresnel law and machine learning method is proposed to identify the layer counts of 2D materials.Three indexes,which are optical contrast,red-green-blue,total color difference,are presented to illustrate and simulate the visibility of 2D materials on Si/SiO_(2) substrate,and the machine learning algorithms,which are k-mean clustering and k-nearest neighbors,are employed to obtain thickness database of 2D material and test the optical images of 2D materials via red-green-blue index.The results show that this method can provide fast,accurate and large-area property of 2D material.With the combination of artificial intelligence and nanoscience,this machine learning assisted method eases the workload and promotes fundamental research of 2D materials.