Adjacent-vertex-distinguishing proper total colouring number of

. Coloring is a fundamental problem in scientific computation and engineering design. In recent years, a variety of colorings frequently appeared and solved many problems in production. For example, adjacent-vertex-distinguishing proper total coloring, adjacent-vertex-distinguishing proper edge coloring, smarandachely-adjacent-vertex-distinguishing proper edge coloring. It is an important also difficult problem to discuss the coloring numbers of a given graph class. And we focus on the adjacent-vertex-distinguishing proper total coloring numbers in this paper. We study the adjacent-vertex-distinguishing proper total coloring numbers of joint graphs


Introduction
Coloring problem in graph theory is one of the most famous NP-complete problems. Four-colorconjecture which is one of the world's three major mathematical conjectures says that each map can only use four colors to dye, and no two adjacent areas dyed same color. In the spring of 1976, with the help of the computer, the four-color-conjecture was proved. The conjecture finally became a theorem. The significance of graph coloring theory is much more than that. Known to all, coloring problems can solve many problems such as scheduling problem, time tabling, transportation, arrangement, circuit design and storage problems.
In recent years, more and more colouring concepts was put forward by experts of graph theory, such as adjacent-vertex-distinguishing proper edge colouring, adjacent-vertex-distinguishing proper total colouring, smarandachely adjacent-vertex-distinguishing proper edge colouring. Symbol  in the paper always denote the maximum degree of the graphs discussed.
Definition 1 [1] A k-proper total colouring for a graph G is a mapping  The number G k | min{ has a k-adjacent-vertex-distinguishing proper total colouring} is called the adjacent -vertex-distinguishing proper total colouring number and denoted by ) (G at  . This concept was put forward by Zhang Zhongfu in [1], and researched by many students and researchers of graph theory. Zhang Zhongfu put forward a conjecture on it such that: Conjecture 1 [1] For every connected graph G with order at least 2, we have 3 In the same time, Zhang Zhongfu put forward an open question in [1] such that: The Conjecture 1 was proved to be valid when the maximum degree of graph is 3 in paper [3]. Huang Danjun give an upper bound of the adjacent-vertex-distinguishing proper total colouring number when the maximum degree of the graph 3  [4].
Determining the adjacent-vertex-distinguishing proper total colouring number of a given type of graphs is a main problem in the question, and common methods that used in colouring problems are such that: giving the specific methods of the colouring; combination analysis methods; probabilistic methods.
Lemma 1 If two arbitrary distinct vertices of maximum degree in G are not adjacent, Proof For every graph G , colours that are used in the vertices and it's adjacent edges are more than 1  , so we get the result such as 1 ) If there are two vertices whose degree are the maximum of the graph who is denoted by u and v , then the cardinal numbers of the colour sets for u and v satisfied that | ( ) | | ( ) | 1 C u C v     , but to be a k-adjacent-vertex-distinguishing proper total colouring, There must be the condition such as In the following paper, we will study the adjacent-vertex-distinguishing proper total colouring of the graph m n KP  which is a joint graph which are jointed by m K and n P . m K denotes graphs with order m and also have no edge. n P denotes the path graphs with order n . That is to say By comparing the colour sets of all vertices for the graph, it is easy can be seen that f is a AVDTC, Listing the colour sets of all vertices for the mapping, we can see that f is a AVDTC, s 5 ) ( By definition of f , then we can get