On the adjacent vertex-distinguishing acyclic edge coloring of some graphs

A proper edge coloring of a graph G is called adjacent vertex-distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the coloring set of edges incident with u is not equal to the coloring set of edges incident with v, where uv∈ E（G）. The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by X＇Aa（G）, is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. If a graph G has an adjacent vertex distinguishing acyclic edge coloring, then G is called adjacent vertex distinguishing acyclic. In this paper, we obtain adjacent vertex-distinguishing acyclic edge coloring of some graphs and put forward some conjectures.

Adjacent Vertex-distinguishing E-total Coloring on Some Join Graphs Cm ∨ Gn

Let G(V, E) be a simple connected graph and k be positive integers. A mapping f from V∪E to {1, 2, ··· , k} is called an adjacent vertex-distinguishing E-total coloring of G(abbreviated to k-AVDETC), if for uv ∈ E(G), we have f(u) ≠ f(v), f(u) ≠ f(uv), f(v) ≠ f(uv), C(u) ≠C(v), where C(u) = {f(u)}∪{f(uv)|uv ∈ E(G)}. The least number of k colors required for which G admits a k-coloring is called the adjacent vertex-distinguishing E-total chromatic number of G is denoted by x^e_(at) (G). In this paper, the adjacent vertexdistinguishing E-total colorings of some join graphs C_m∨G_n are obtained, where G_n is one of a star S_n , a fan F_n , a wheel W_n and a complete graph K_n . As a consequence, the adjacent vertex-distinguishing E-total chromatic numbers of C_m∨G_n are confirmed.

Adjacent Vertex Distinguishing I-total Coloring of Outerplanar Graphs

Let G be a simple graph with no isolated edge. An/-total coloring of a graphG is a mapping Ф ： V（G） U E（G） → （1, 2,…… , k） such that no adjacent vertices receive thesame color and no adjacent edges receive the same color. An/-total coloring of a graph G issaid to be adjacent vertex distinguishing if for any pair of adjacent vertices u and v of G, wehave CФ（u） ≠ CФ（v）, where CФ（u） denotes the set of colors of u and its incident edges. Theminimum number of colors required for an adjacent vertex distinguishing I-total coloring of GG is called the adjacent vertex distinguishing I-total chromatic number, denoted by Xat（G）.In this paper, we characterize the adjacent vertex distinguishing I-total chromatic numberof outerplanar graphs.

I-Total Coloring and VI-Total Coloring of mC_(4) Vertex-Distinguished by Multiple Sets

We give the optimal I-(VI-)total colorings of mC_(4)which are vertex-distinguished by multiple sets by the use of the method of constructing a matrix whose entries are the suitable multiple sets or empty sets and the method of distributing color set in advance.Thereby we obtain I-(VI-)total chromatic numbers of mC_(4)which are vertex-distinguished by multiple sets.

On the Adjacent Vertex-distinguishing Equitable Edge Coloring of Graphs

Let G（V, E） be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if （1） C（u）≠C（v） for uv ∈ E（G）, where C（u） = {f（uv）｜uv ∈ E}, and （2） for any i, j = 1, 2,… k, we have ｜｜Ei｜ ｜Ej｜｜ ≤ 1, where Ei = {e｜e ∈ E（G） and f（e） = i}. χáve （G） = min{k｜ there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the χ áve （G） of some special graphs and present a conjecture.