Zhongfu Zhang

The Smarandachely Adjacent-Vertex Total Coloring of Three Classes of 3-Regular Halin Graphs

The Smarandachely adjacent-vertex total chromatic number of graph G is the smallest positive integer k for which G has a proper total k-coloring such that for any pair of adjacent vertices u, v, the set of colors appearing at vertex u and its incident edges is not a subset of the set of colors appearing at vertex v and its incident edges. This paper,we obtain the Smarandachely adjacent-vertex total chromatic numbers of three particular classes of 3-regular Halin graphs. I. INTRODUCTION Some new colorings of graphs are produced from applied areas of computer science, information science and light transmission. They are the vertex distinguishing edge coloring (1), the adjacent vertex distinguishing edge coloring (2, 3, 4), the adjacent vertex distinguishing total coloring (5,6,7,8), the D(β)-vertex-distinguishing total colorings (9), the adjacent- vertex strongly distinguishing total coloring of graphs (10), vertex-distinguishing total coloring of graphs (11) and the relation of total chromatic number with adjacent strong edge chromatic number of regular graph (12). Recently, Zhang et al. (11) propose the Smarandachely adjacent vertex total color- ing on graphs. In this paper, We obtain the Smarandachely adjacent-vertex total chromatic numbers of three particular classes of 3-regular Halin graphs. Definition 1.1. (10) A proper total k-coloring of G is a mapping π : V (G) ∪ E(G) →{ 1, 2 ,...,k } such that two adjacent or incident elements of V (G) ∪ E(G) are assigned distinct colors. Let C(u )= {π(u) }∪{ π(uv ): uv ∈ E(G)} for any vertex u ∈ V (G) .I fC(u) �= C(v) for uv ∈ E(G) ,w e say that π is an adjacent-vertex distinguishing total k-coloring (a k-AVDTC) of G. The minimum number such that G admits a k-AVDTC, denoted by χat(G), is called the adjacent-vertex distinguishing total chromatic number of G. Conjecture 1.1. (10) For a simple graph G, then χat(G) ≤ Δ(G )+3 . Definition 1.2. (14) A Smarandachely adjacent-vertex total coloring (k-SAVTC) of a graph G is a proper total k-coloring π of G such that |C(u) \ C(v) |≥ 1 and |C(v) \ C(u) |≥ 1 for all uv ∈ E(G). The minimum number such that G admits a k-SAVTC, denoted by χsat(G), is called the Smarandachely total chromatic number of G. Conjecture 1.2. (14) Let G be a connected graph of orders at least 3, then Δ(G )+2 ≤ χsat(G) ≤ Δ(G )+3 . Conjecture 1.3. (14) Let G be a r-regular graph but not a complete graph of odd order, then χsat(G )= r +2 . Definition 1.3. (15) For a 3-connected planar graph G(V, E, F), if all edges on the boundary of one face f0 of the set F are removed, it becomes a tree, and the degree of each vertex of V (f0) is three, then graph G is called a Halin graph, the vertices on V (f0) are called exterior vertices of G, and the others interior vertices of G. Definition 1.4. The center of a graph G is a vertex, which is passed by all the longest path in graph G. Definition1.5. For a 3−regular Halin graph H,if V (H )= {ui ,v i ,w j|j =1 , 2 ,i =1 , 2, ··· ,n }; E(H )=

The Smarandachely adjacent-vertex distinguishing total coloring of two kind of 3-regular graphs

The Smarandachely adjacent-vertex distinguishing total coloring of graphs is a proper k-total coloring such that every adjacent vertex coloring set not embrace each other, the minimal number k is denoted the Smarandachely adjacent-vertex distinguishing total coloring chromatic number of graphs. Where the coloring set include the colors of all edges incident to the vertex plus the color of it. In this paper, we construct two kind of 3-regular graph Rn3 and S4n3, and obtain the Smarandachely adjacent-vertex distinguishing total coloring chromatic number of it.

Incidence Adjacent Vertex-Distinguishing Total Coloring of Graphs

Let G(V, E) be a simple graph,k is a positive integer.f is a mapping from V (G) ∪ E(G) to {1, 2, ··· ,k } such that ∀uv ∈ E(G),then f (u) �= f (v); ∀uv, vw ∈ E(G) ,u �= w, f(uv) �= f (vw); ∀uv ∈ E(G) ,C (u) �= C(v),we say that f is the incidence-adjacent vertex distinguishing total coloring of G.The minimum number of k is called the incidence-adjacent vertex distinguishing total chromatic number of G.Where C(u )= {f (u) }∪{ f (uv)|uv ∈ E(G)}. In this paper, we discuss some graphs whose incidenceadjacent vertex distinguishing total chromatic number is just Δ,Δ+1,Δ+2, and present a conjecture that the incidenceadjacent vertex distinguishing total chromatic number of a graph is no more than Δ +2 .

Adjacent Vertex Reducible Vertex-Total Coloring of Graphs

Let G =( V, E) be a simple graph,k (1 ≤ k ≤ Δ(G )+1 ) is a positive integer. f is a mapping from V (G) ∪ E(G) to {1, 2, ··· ,k } such that ∀uv ∈ E(G),f (u) �= f (v) and C(u )= C(v) if d(u )= d(v),we say that f is the adjacent vertex reducible vertex-total coloring of G. The maximum number of k is called the adjacent vertex reducible vertextotal chromatic number of G, simply denoted by χavrvt(G). Where C(u )= {f (u)|u ∈ V (G) }∪{ f (uv)|uv ∈ E(G)}. In this paper,the adjacent vertex reducible vertex-total chromatic number of some special graphs are given.

Adjacent Vertex Reducible Edge-Total Coloring of Graphs

Let G(V,E) be a simple graph,k (1 k + 1) is a positive integer. f is a mapping from V (G) [ E(G) to {1,2,...,k} such that 8uv,uw 2 E(G),v 6= w,f(uv) 6= f(uw);8uv 2 E(G) , if d(u) = d(v)then C(u) = C(v);we say that f is the adjacent vertex reducible edge-total coloring of G. The maximum number of k is called the adjacent vertex reducible edge-total chromatic number of G, simply denoted by avret(G). Where C(u) = {f(u)|u 2 V (G)} [ {f(uv)|uv 2 E(G)}. In this paper,the adjacent vertex reducible edge-total chromatic number of some special graphs.

EDGE-FACE CHROMATIC NUMBER OF 2-CONNECTED PLANE GRAPHS WITH HIGH MAXIMUM DEGREE*

The edge-face chromatic number χef(G) of a plane graph G is the least number of colors assigned to the edges and faces such that every adjacent or incident pair of them receives different colors. In this article, the authors prove that every 2-connected plane graph G with Δ(G)≥∣G∣ − 2 ≥ 9 has χef(G) = Δ(G).