General Vertex-Distinguishing Total Coloring of Graphs

The general vertex-distinguishing total chromatic number of a graph G is the minimum integer k, for which the vertices and edges ofG are colored using k colors such that any two vertices have distinct sets of colors of them and their incident edges. In this paper, we figure out the exact value of this chromatic number of some special graphs and propose a conjecture on the upper bound of this chromatic number.


Introduction
All graphs considered in this paper are simple and finite.For a graph , we denote by (), (), Δ(), and () the sets of vertices, edges, maximum degree, and minimum degree of , respectively.For a vertex V of ,   (V) is the degree of V in .For any   ⊆ (), we use [  ] to denote the subgraph induced by   .For any undefined terms, the reader is referred to the book [1].
The coloring problem of graphs is one of the classical research areas in graph theory.It has been widely applied to various fields, such as large scheduling [2], assignment of radio frequency [3], and separating combustible chemical combinations [4].Due to its extensive application, many new variants of colorings have been studied [5].
Recall that a -edge coloring of a graph  is a mapping : () → , where  is a set of  colors.An edge coloring is proper if adjacent edges receive distinct colors.In 1985, Harary and Plantholt [6] first considered pointdistinguishing chromatic index, which is a variant of edge coloring.After that, many other variants of edge coloring were introduced, such as vertex-distinguishing proper edge coloring [7], adjacent vertex-distinguishing edge coloring [8], and general adjacent vertex-distinguishing edge coloring [9].
A total -coloring of a graph  is a coloring of ()∪() using  colors.A total -coloring is proper if no two adjacent or incident elements receive the same color.The minimum number of colors required for a proper total coloring of  is called the total chromatic number of  and is denoted by   ().Behzad [10] and Vizing [11] independently made the conjecture that, for any graph ,   () ≤ () + 2. ( This is known as the total coloring conjecture () and is still unproven.
Let  be a total -coloring of .The total color set (with respect to ) of a vertex V ∈ () is the set, denoted by   (V), of colors of V and its incident edges.We denote by C  () the set of total color sets of all vertices of .Furthermore, let  be a subset of () ∪ (); we use   () to denote the set of colors of elements of .
Like edge coloring, total coloring also has some variants.In 2005, Zhang et al. [12] added a restriction to the definition of total coloring and proposed a new type of coloring defined as follows.
Definition 1.Let  be a proper total -coloring of a graph .If, for all , V ∈ (),   () ̸ =   (V), then  is called an adjacent vertex-distinguishing total -coloring of , or a -AVDTC of  for short.The minimum number  for which  has a -AVDTC is the adjacent vertex-distinguishing total chromatic number of , denoted by   ().
For a -AVDTC of a graph , if   () ̸ =   (V) is required for any two distinct vertices , V, then  is called a vertex-distinguishing total -coloring of , abbreviated as -VDTC.The minimum number  such that  has a -VDTC is called the vertex-distinguishing total chromatic number, denoted by  V () [24].Zhang et al. conjectured in [24] that, for any graph , it follows that where   () = min{ | (  +1 ) ≥   ,  ≤  ≤ Δ}.In this paper, we introduce a variant of vertexdistinguishing total coloring of a graph , which relaxes the restriction that the coloring is proper.We now present the detailed definition as follows.
Definition 2. Let  be a graph and  be a positive integer.A total coloring  of  using  colors is called a general vertex-distinguishing total -coloring of  (or -GVDTC of  briefly) if, for all , V ∈ (),   () ̸ =   (V).The minimum number  for which  has a -GVDTC is the general vertexdistinguishing total chromatic number, denoted by  V ().
It is evident that  V () does exist for any graph .In this paper, we study the general vertex-distinguishing total coloring of some special classes of graphs and obtain the exact value of the general vertex-distinguishing total chromatic number of these graphs.Furthermore, we propose a conjecture on the upper bound of general vertex-distinguishing total chromatic number of a graph.

Main Results
We first present a trivial lower bound on the general vertexdistinguishing total chromatic number of a graph.
We now turn to investigating the general vertexdistinguishing total chromatic number of an -vertex path.
According to Theorem 5, we have the same conclusion on cycles.Let
and let  be a  * -GVETC of   , constructed by the method of Theorem 5. Then we can extend  to a  * -GVETC of   by assigning color 1 to edge V 1 V  .So, the conclusion holds.
For two vertices , V of a graph , to identify these two vertices is to replace them by a single vertex (denoted by -V in this paper) incident to all the edges which were incident in  to either  or V.The resulting graph is denoted by /{, V}.
In the above, we construct a -GVDTC of a graph  by extending a -GVDTC of graph   , where   is the resulting graph of identifying two vertices of degree more than 1 in .But this method does not always work.For instance, the graph /{, V} shown in Figure 1(b) has a 4-GVDTC, but the graph  shown in Figure 1(a) does not contain any 4-GVDTC.So any 4-GVDTC of /{, V} can not be extended to a 4-GVDTC of .
In the following we are devoted to the study of the general vertex-distinguishing chromatic number of fan graph   , wheel graph   , and complete graph   .Let ,  be two graphs such that () ∩ () = 0.The join + of  and  is a graph with vertex set ( + ) = () ∪ () and edge set ( + ) = () ∪ () ∪ {V |  ∈ (), V ∈ ()}.A fan graph   is defined as the join of a path of  vertices and an isolated vertex.A wheel graph   is defined as the join of a cycle of  vertices and an isolated vertex.